For years, Exxon Mobil Corp. didn’t have to pay much attention to investors because of its gargantuan profits. Yet on a Friday night in January, Exxon Chief Executive Darren Woods was defending the company during a video call to an investor owning about 0.02% of the oil giant’s stock.

Tech investor Chris James’s Engine No. 1 had launched an activist campaign against Exxon in December, calling the company a fossil-fuel dinosaur that lacked a coherent plan for surviving a global transition to cleaner energy sources. On the call, Charlie Penner, a hedge-fund veteran helping lead the Engine No. 1 campaign, pressed Mr. Woods to commit to steering Exxon to carbon neutrality, effectively bringing its emissions to zero—both from the company and its products—by 2050.

Mr. Woods refused, arguing that oil companies making such pledges had no real plans to achieve them. “They weren’t interested in having a conversation,” he said in a recent interview. “Frankly, they didn’t have a plan.”

Messrs. Woods, James and Penner failed to come to any agreement in what ended in a contentious exchange, people familiar with the matter said.

Since January, Engine No. 1’s bid for four seats on Exxon’s board has turned into one of the most expensive proxy fights ever. Exxon has spent at least $35 million, and Engine No. 1 has spent $30 million, regulatory filings show, in an increasingly pitched battle to persuade shareholders voting Wednesday at the company’s annual meeting.

The activists got a boost Tuesday when BlackRock Inc., one of Exxon’s largest shareholders, decided to back three of Engine No. 1’s four board nominees, according to people familiar with the matter. Engine No. 1 hasn’t called for Mr. Woods’s removal, but many view the vote as a referendum on his performance, and the outcome could affect his ability to execute his strategy.

The fight demonstrates the challenge facing Mr. Woods: He is defending Exxon’s pumping of oil and gas just as finance is moving decisively toward funding a future based on renewable resources.

Darren Woods, the chairman and chief executive of Exxon Mobil Corp.

Darren Woods, the chairman and chief executive of Exxon Mobil Corp.

Photo: Richard Drew/Associated Press

“One of the things I’ve learned in this job and particularly with this activist campaign, is the need and the opportunity for us to do a better job of explaining what we’re doing,” Mr. Woods said. “We are certainly a large, iconic U.S. company. We are associated with the industry…that brings a spotlight.”

Less than a decade ago, Exxon was the largest U.S. company by market capitalization, and the idea of an activist campaign challenging its leadership would have been unthinkable. Former Exxon CEO Lee Raymond sometimes publicly belittled questions by bank analysts on the company’s strategy as stupid, once likening some analysts to “Mickey Mouse” and “Goofy.”

The underpinning of that attitude has largely crumbled. Exxon last year posted a $22 billion loss when the pandemic crushed fuel demand and upended what turned out to be an ill-timed plan by Mr. Woods to substantially increase spending to boost oil and gas production. The loss has given momentum to the campaign by Engine No. 1, which has sought to capitalize on investors’ fears about years of shrinking profits and concerns about the company’s future, as governments increase regulations to address climate change.

So far, the country’s three largest pension funds—the California State Teachers’ Retirement System, California Public Employees’ Retirement System and the New York State Common Retirement Fund—have said they would support Engine No. 1’s candidates.

A spokeswoman for Engine No. 1 disputed that it was unwilling to have a dialogue with Exxon. The fund has accused Exxon’s board of presiding over the company’s demise and argued its own candidates have energy experience, unlike many current directors.

“Engine No. 1’s plan is to add four directors who have all profitably looked around corners in the energy industry and can help management achieve long-term success in a rapidly changing world,” she said.

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On Monday, Exxon wrote shareholders pledging to appoint two directors with energy industry and climate experience within the next 12 months, part of a monthslong charm offensive. Since December, Exxon has spent millions of dollars on advertising and mailed shareholder solicitations. It has appointed three new directors and announced changes long sought by some investors, including creating a business unit for carbon emissions-reducing technologies and disclosing for the first time the emissions from Exxon products.

“At a certain point you have to decide, ‘If you’re not with me, then you’re not with me,’ ” Mr. Woods said in a 2019 interview at his alma mater, Texas A&M University.

Some investors fear that attitude has made the oil giant slow to react to a changing energy landscape.

An entrance to the Exxon Mobil Chemical complex in Baytown, Texas.

An entrance to the Exxon Mobil Chemical complex in Baytown, Texas.

Photo: Brandon Thibodeaux for The Wall Street Journal

In a 2019 meeting of business leaders with Pope Francis at the Vatican, the pontiff implored the CEOs and executives of Exxon, BP PLC, Royal Dutch Shell PLC, BlackRock, State Street Corp. and others to accept moral responsibility to help clean up the planet. Mr. Woods acknowledged the need to address climate change, but he said Exxon had to do so in a way that satisfied its obligation to generate returns for shareholders. His response frustrated his peers at the meeting, which was perceived as tone deaf, people familiar with the matter said.

Stephen Arbogast, a 32-year Exxon executive who is now director of the Energy Center at the University of North Carolina-Chapel Hill, said the role of big oil CEO has evolved, and now requires communicating the company’s future in a low-carbon world to a broader audience.

“Many of these audiences believe that climate is an existential threat, and that oil and gas need to be sunset industries,” Mr. Arbogast said.

Up the ranks

Mr. Woods said he didn’t seek out the job of Exxon’s chief executive. When his predecessor, Rex Tillerson, offered him the position in 2016, Mr. Woods told him he didn’t want it.

“Frankly I’m not interested in the limelight,” Mr. Woods said, adding that he has learned to deal with it.

Whatever his ambition, current and former colleagues of Mr. Woods say he was marked to climb the ladder at Exxon from early on.

The son of a military supplier, Mr. Woods grew up living at or near U.S. bases around the world. He was an offensive lineman on his Texas high-school football team. He followed his high-school sweetheart, now his wife, to Texas A&M and earned a degree in electrical engineering. He received an M.B.A. at Northwestern University and joined Exxon in 1992.

Mr. Woods eventually agreed to lead the company. The board selected him, in part, because he had climbed the ranks of Exxon’s refining division, where profits are made squeezing pennies from every barrel. He also spoke the language of Wall Street, having spent time in Exxon’s investor relations division, according to the people. The board wanted a break from Mr. Tillerson, a plain-spoken Texan oilman who had spent billions of dollars in pursuit of high-price oil projects that haven’t worked out, these people said.

Current and former Exxon executives describe Mr. Woods as demanding. Before signing off on investment ideas, he subjected employees to marathon review sessions sometimes lasting more than a day. He employed Exxon’s practice of randomly assigning executives to blue and red teams, one to tear down an idea, the other to defend it.

Big bets

Some former Exxon executives said Mr. Tillerson left Mr. Woods a bad hand.

In 2016, Mr. Tillerson’s final full year, Exxon’s return on capital was 1.3%, down from 26% in 2005, during Mr. Raymond’s last year, according S&P Global Market Intelligence. Exxon had nearly $21 billion in its war chest when Mr. Tillerson took over, and had more than $39 billion in net debt when he left.

Some of Mr. Tillerson’s biggest bets, including investments in Canadian oil sands and U.S. shale gas, came during a period of high commodity prices, and they failed to generate strong profits when prices fell. Exxon also had to exit some joint ventures with state companies in Russia as a result of Western sanctions. It wrote down U.S. shale gas, Canadian oil sands and other properties by $19 billion this year. Mr. Tillerson didn’t respond to requests for comment.

Rex Tillerson, former chief executive officer of Exxon Mobil Corp., in 2019.

Rex Tillerson, former chief executive officer of Exxon Mobil Corp., in 2019.

Photo: Jeenah Moon/Bloomberg News

Despite that, Mr. Woods continued Mr. Tillerson’s big spending, laying out a plan in 2018 to invest $230 billion to pump an additional one million barrels of oil and gas a day by 2025. He sought to differentiate his plan, saying it was a return to Exxon’s old playbook: large, disciplined investments on prospects that can make money at low oil prices.

Even before the pandemic, the strategy wasn’t yielding immediate results. Production was roughly flat through 2019, Exxon’s return on capital that year was just 3% and its net debt ballooned to nearly $50 billion, according to S&P Global Market Intelligence.

The spread of Covid-19 wrecked Mr. Woods’s plans. As countries world-wide imposed quarantines, fossil fuel demand plummeted. Exxon responded by slashing its planned capital spending between 2021 and 2025 by about a third and said it would cut as much as 15% of its global workforce.

Its $22 billion annual loss last year was the first ever. In August, the company was removed from the Dow Jones Industrial Average, after nearly a century on the index.

Exxon’s lead director, Merck & Co. Chief Executive Kenneth Frazier, defended Mr. Woods’s performance in an interview, saying he had to make difficult decisions to pull the plug on projects and carefully choose among new ones as soon he became CEO.

“Darren, in his tenure, has had the wind in his face,” said Mr. Frazier, who will retire as Merck’s chief executive in June. “CEOs should be judged, in part, by how they develop and how they execute plans at the bottom of the cycle.”

Exxon’s lead director Kenneth Frazier, the chairman and chief executive of Merck & Co., joined by President Joe Biden in March.

Exxon’s lead director Kenneth Frazier, the chairman and chief executive of Merck & Co., joined by President Joe Biden in March.

Photo: Andrew Harnik/Associated Press

Some of Mr. Woods’s biggest investments could turn out to be profitable for Exxon as oil demand recovers and Exxon’s peers, including BP PLC and Royal Dutch Shell PLC, divert money from oil production to renewables. In Guyana, where Exxon made one of the largest oil discoveries in years during Mr. Tillerson’s tenure, the company could see a return on investment of 20% or more, analysts said, far surpassing returns from renewable projects.

Still, many investors are growing increasingly disaffected. Exxon’s stock has recovered after plummeting during the pandemic, yet still trades at only about 1.5 times its book value, down from about 3.5 times book value shortly after Mr. Raymond left.

Tough sell

Engine No. 1 is banking on that dissatisfaction. It has hammered home Exxon’s financial underperformance to investors, while also challenging Exxon’s refusal to entertain the idea that fossil fuel demand may decline faster than anticipated.

Mr. Woods has acknowledged the contribution of fossil fuels to climate change, unlike Mr. Raymond, and said Exxon can help reduce emissions while staying committed to oil and gas. In March, he unveiled a strategy that would maintain Exxon as the largest Western oil producer. That has proven a tough sell with some investors.

Large money managers, like BlackRock, are also under pressure to exert influence on their portfolio companies to do more about climate change. BlackRock, State Street and Vanguard‌collectively‌own‌more‌than 20% of Exxon’s shares and could tip the scale on Wednesday’s vote. All three have signed a pledge supporting goals to reach net zero carbon emissions by 2050 or sooner. The International Energy Agency said investment in new fossil-fuel projects must stop immediately if the world was going to achieve that.

Legal & General, Britain’s biggest asset manager, which owns about $1 billion in Exxon shares, said this month it couldn’t support Mr. Woods’s strategy and would vote for Engine No. 1’s slate. John Hoeppner, who heads its U.S. Sustainable Investments, said his team had met with Exxon representatives several times in recent months and noticed a more conciliatory tone, but that Exxon had not offered substantive proposals to address the firm’s concerns.

“It was like, ‘We heard you, investment community, we listened and delivered, now let’s move on,’ ’’ he said. “They don’t recognize there is a structural challenge.”

The Exxon Mobil Chemical complex, in Baytown, Texas last year.

The Exxon Mobil Chemical complex, in Baytown, Texas last year.

Photo: Brandon Thibodeaux for The Wall Street Journal

Other investors said they had spoken directly to Mr. Woods in recent months and that he had improved investor outreach since becoming chief executive. But, they said, Mr.‌Woods‌is‌unwavering about‌his views‌that demand for oil and gas will increase in coming years and unwilling to entertain suggestions that Exxon‌ought‌to hedge‌its‌bets.

Engine No. 1 has said Exxon’s moves since the proxy fight began are inadequate. It has portrayed Exxon’s low-carbon unit as an exercise in greenwashing, more about public relations than investment. The fund said Exxon refused to meet with its candidates. Exxon said it reviewed Engine No. 1’s candidates and determined they didn’t meet the board’s standards.

Despite the contention, Engine No. 1 said it wasn’t calling for Exxon to unwind its oil and gas business but to gradually diversify itself to be ready for a world that will need less oil and gas. Committing itself to cutting carbon emissions to zero isn’t only a benefit to society but a business imperative for Exxon in a changing energy market, it said.

Mr. Woods is adamant that Exxon will be a leader in the energy transition. Exxon was making substantial investments to reduce carbon emissions, he said, while meeting the world’s need for oil and gas.

“Whatever the future shapes up to be,” Mr. Woods said, “we will be more valuable. That is the objective that we’re working for.”

—Cara Lombardo contributed to this article.

Write to Christopher M. Matthews at christopher.matthews@wsj.com