Tesla Inc. Chief Executive Elon Musk, who has repeatedly played down the risk of the coronavirus since early in the pandemic, says he tested positive for Covid-19 and sought to raise doubts about the validity of such testing more broadly.

Mr. Musk said on Twitter a Thursday night saliva sample came back positive for the virus. He took the test after earlier rapid-response tests he had taken returned both positive and negative results.

The initial four Covid-19 results Mr. Musk said he received came from rapid-response tests. They aren’t considered as accurate as the later PCR, or polymerase chain reaction, tests that typically involve a nasal or throat swab.

Mr. Musk said Saturday he was getting another test with results expected Sunday. “Almost no symptoms today, so hopefully better results,” he tweeted.

After the earlier mixed outcome of the rapid-response tests, Mr. Musk late Thursday tweeted “Something extremely bogus is going on,” later adding, “If it’s happening to me, it’s happening to others.”

His comments about the results met with a chorus of Twitter followers who shared skepticism about the tests, alleging high false-positive rates and widespread misdiagnosis. Among those making the claims were groups and individuals who have publicly opposed the lockdowns and other restrictions mandated by governments to try to stem the spread of the virus.

Health officials in the U.S. are reporting higher infections, as well as more hospitalizations due to Covid-19 that rose to a record 68,516 Friday, according to the Covid Tracking Project.

The symptoms Mr. Musk initially said he had experienced were mild “sniffles & cough & slight fever past few days.”

If a diagnosis is confirmed, Mr. Musk would become one of several CEOs to reveal they contracted the virus. Morgan Stanley CEO James Gorman, NBCUniversal CEO Jeff Shell and Hewlett-Packard Enterprise Co. CEO Antonio Neri revealed positive diagnoses earlier this year.

Howard Willard, former CEO of Altria Group Inc., took a temporary medical leave after a positive diagnosis in March and announced his retirement in April after a rocky two-year tenure leading the Marlboro maker.

Through the pandemic, management teams have worked to fortify succession plans and review backup operating plans when critical employees fall ill.

In March, Mr. Musk said there would likely be close to zero new Covid-19 cases in the U.S. by the end of April. The outspoken CEO also said in March: “My guess is that the panic will cause more harm than the virus, if that hasn’t happened already.”

Mr. Musk disclosed the test results ahead of a planned rocket launch by Space Exploration Technologies Corp., where he also is CEO. The launch from the National Aeronautics and Space Administration’s Kennedy Space Center in Florida is scheduled to carry four astronauts to the International Space Station—the launch was scheduled for Saturday, but got delayed as a result of onshore winds and for booster-recovery operations.

NASA Administrator Jim Bridenstine on Friday said it was the agency’s policy that when someone tests positive, they self-isolate, and that it was looking to SpaceX to handle any appropriate contact tracing. “If there are adjustments that need to be made, we will make them,” he said.

The astronauts scheduled to launch on the Falcon 9 rocket have been quarantining and, Mr. Bridenstine said, he wasn’t aware of any contact between them and Mr. Musk. The crew, he said, “should be in good shape.”

Among the questions Mr. Musk on Twitter raised about Covid-19 testing was the level of false-positive rates for PCR tests. “The guy is spreading more misinformation on Covid than almost anyone else out there,” said Ashish Jha, professor of health services and dean of the Brown University School of Public Health. “Things are horrible and he is actively using his platform to say things that will lead to more infections and deaths. It is deeply irresponsible and he should be better than this.”

Mr. Jha said one of Mr. Musk’s deputies contacted him early in the pandemic and asked him to make public statements asserting that a large number of false positive tests made the pandemic appear much worse than it was. Mr. Jha declined, he said, because “the idea that there are all these false positives with PCR tests is junk.”

Mr. Musk and Tesla didn’t respond to a request for comment.

On Thursday, Alameda County, near San Francisco—where Tesla’s lone U.S. car factory is based—warned of rising coronavirus cases and potential new restrictions to combat the disease. County health officials earlier this year ordered Tesla to temporarily close the car plant.

On an earnings call in April, with the U.S. plant shut, Mr. Musk railed against local shelter-in-place restrictions aimed at slowing the spread of the virus.

“Give people back their goddamn freedom,” he said.

When Tesla reopened the Fremont plant in May, it had put in place safety protocols that county officials inspected. Workers have reported instances of Covid-19 cases among workers at the facility, though neither Tesla nor local authorities have commented on the scale of infection among the vehicle maker’s staff.

The response to the pandemic has become a polarized issue in America. Mr. Musk retweeted several comments that sought to raise questions about the accuracy of tests.

From the Archives

Aerial footage showed forklifts in action, people walking and cars entering Tesla's car plant in Fremont, Calif., on Monday, as Elon Musk said the electric-vehicle maker would defy a county order and resume production. Photo: David Paul Morris/Bloomberg News (Originally Published May 12, 2020)

The test, which Mr. Musk said was from Becton, Dickinson & Co., is one of several authorized antigen tests, which search for virus proteins in patient samples and can deliver a result in about 15 minutes. The tests tend to be less precise than laboratory-based PCR tests. But they are good at identifying those that have higher viral loads and are likely most infectious, public-health experts say.

“We are aware of the tweet and are reaching out to learn more, consistent with our quality management process,” a spokeswoman for Becton Dickinson said. “We stand by the quality, utility and science of our system and assay. There are many factors that could lead to a discordant result, including a low viral load.”

False positives and false negatives can occur with any clinical test. False negative results, or failing to pick up a present infection, are more common with antigen tests, but false positives can also occur. Public-health authorities sometimes recommend a confirmatory PCR test and say that test results should be looked at in conjunction with other pieces of information, such as symptoms and potential exposure.

Becton Dickinson in September said it was investigating reports from nursing homes that federally provided rapid coronavirus testing equipment from the company was producing false-positive results in some cases.

It couldn’t be determined why Mr. Musk took four rapid tests or how he had access to so many when many members of the public at times have struggled to get timely testing.

Despite the turmoil of recent months and the wider global economic slowdown, Tesla has been navigating the health crisis with little apparent impact. Vehicles deliveries in the second quarter fell compared with the year-ago period.

The pandemic at one point threatened to derail Mr. Musk’s plan to boost Tesla deliveries by about 36% this year with the closure of the plant. Mr. Musk fought to reopen the factory, and Tesla last month said its goal of delivering more than 500,000 vehicles this year could still be attainable. The company is on track to post its first full-year profit in 2020 despite the pandemic, according to analyst estimates.

Tesla shares were down about 0.79% Friday afternoon as the broader market advanced.

—Bowdeya Tweh and Brianna Abbott contributed to this article.

Corrections & Amplifications
Both Mr. Musk’s Covid-19 tests and Alameda County’s advisory occurred Thursday. An earlier version of this article incorrectly said they occurred Wednesday. (Corrected on Nov. 13)

Write to Robert Wall at robert.wall@wsj.com and Heather Somerville at Heather.Somerville@wsj.com