A jury rejected Sarah Palin’s libel suit against The New York Times on Tuesday, a day after the judge said he would dismiss the case if the jury ruled in her favor because her legal team had failed to provide sufficient evidence that she had been defamed by a 2017 editorial erroneously linking her to a mass shooting.

The jury’s verdict, and the judge’s decision, served as a validation of the longstanding legal precedent that considers an occasional mistake by the media a necessary cost of discourse in a free society. And it came as those who want to see journalists pay a steeper legal cost for getting something wrong are pushing the Supreme Court to reconsider the issue.

In absolving the The Times of liability, the jury concluded that the newspaper and its former opinion editor, James Bennet, had not acted with the level of recklessness and ill intent required to meet the high constitutional burden for public figures who claim defamation.

Ms. Palin is expected to appeal, but appeals courts tend to be deferential to decisions made by juries. Even if she does not succeed, those who want to revisit the current standard of libel, which was set by the 1964 case The New York Times Company v. Sullivan, will continue their push to find a case to challenge the established precedent, legal experts said.

It requires the votes of four justices for the Supreme Court to take a case. Given critical comments that justices like Clarence Thomas and Neil M. Gorsuch have made about the Sullivan ruling, that may be easier to do if the court is presented with the right case than legal scholars would have expected just a short time ago.

“I don’t think it’s a big effort to come up with the three or four extra votes here to revisit,” said David A. Logan, a professor at Roger Williams University School of Law who has argued that the Sullivan standard is too broad.

The fact that this bedrock principle of First Amendment law could be up for review is a surprising development, Mr. Logan added. The skepticism about Sullivan has come mostly, but not exclusively, from the political right. “Two years ago, we would not be having this conversation,” Mr. Logan said.

The Palin suit set up a high-stakes test of the law involving press freedoms. Lawyers for Ms. Palin, the former governor of Alaska and the 2008 Republican vice-presidential nominee, argued that the legal protections in place for half a century to shield journalists from liability are overly broad relics of a pre-internet era. Under the Sullivan standards, a public figure like Ms. Palin has to prove that a news outlet acted with “actual malice” in publishing false information, meaning it displayed a reckless disregard for the truth or knew the information was false.

The Times, which acknowledged and corrected the error in question soon after it was published, has not lost a libel case in an American courtroom in at least 50 years.

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Meredith Kopit Levien, the chief executive of the Times Company, said on Tuesday that the verdict marked “a good day for journalism” and “a good day for the free press.”

Ms. Palin told reporters outside the courthouse that she was “disappointed” in the verdict. Asked if she had a message for The Times, she said: “Tell the truth. It’s as simple as that.”

The jurors made no immediate comment after the verdict.

Ms. Palin’s lawyers may get another chance to make their case for why press protections should be pared back. Legal experts said one avenue for asking an appeals court to reconsider the case was to argue that the courts should revisit the broad definition of a public figure in current law, which makes it extremely difficult for people who have even modest public profiles to successfully sue.

But ultimately, her appeal may never reach the Supreme Court. First Amendment scholars said the appeals court may view her case skeptically given that both the judge and the jury decided she had not met her burden of proof. And the justices may find it more palatable to take a case involving a less politically charged public figure.

“There is unease on the court about The New York Times v. Sullivan test,” said Eugene Volokh, who teaches First Amendment law at the University of California, Los Angeles. But Mr. Volokh said he believed there was little chance that Ms. Palin’s case would end up being the vehicle for any effort to roll it back because the jury did not believe her argument that Mr. Bennet was so blinded by his disgust for her that he knew the editorial contained material that was likely false and overlooked evidence to the contrary.

“It sounds like the jury didn’t buy that,” Mr. Volokh said. “And it becomes very difficult to argue to the court, ‘Well, they thought it was an innocent mistake, but they were wrong.’”

Ms. Palin’s suit claimed that The Times defamed her with an editorial that incorrectly asserted a link between her political rhetoric and a mass shooting near Tucson, Ariz., in 2011 that left six people dead and 14 wounded, including Gabrielle Giffords, then a Democratic member of Congress. Ms. Giffords’s district had been one of 20 singled out underneath digitized cross hairs on a map circulated by Ms. Palin’s political action committee. There was no evidence the shooter had seen or was motivated by the map.

The editorial was published on June 14, 2017, the same day a gunman opened fire at a baseball field in Virginia where Republican congressmen were practicing, injuring several people, including Representative Steve Scalise of Louisiana. The headline was “America’s Lethal Politics,” and the editorial asked whether the Virginia shooting was evidence of how vicious American politics had become. The Times corrected the editorial the morning after it was published after readers pointed out the mistake.

On the witness stand, Mr. Bennet, who inserted the erroneous wording into the article, testified that the incident had left him racked with guilt and that he had thought about it almost every day since. “It was just a terrible mistake,” he said.

Throughout the trial, Ms. Palin’s lawyers attempted to convince the jury that Mr. Bennet had acted out of animus toward her and, regardless of any contrition he later showed, made the error through a combination of carelessness and a willful overlooking of facts. Often, the evidence they produced in internal Times emails and the answers they elicited during a week of testimony painted an unflattering picture of the inner workings of the news organization.

The Times journalists involved in writing, editing and fact-checking the editorial testified about lapses and oversights they regretted. The original writer of the article said on the witness stand, for example, that she had not read very carefully the version Mr. Bennet rewrote. A fact checker said she had overlooked the line about political incitement that triggered Ms. Palin’s suit.

Times lawyers pointed to a series of steps taken by Mr. Bennet and others that they said demonstrated how seriously The Times had taken the issue upon learning about the mistake — including an email Mr. Bennet sent at 5 a.m. the morning after the editorial was published seeking to resolve the issue as soon as possible and the fact that once The Times published a correction it drew attention to it on social media.

The ruling on Monday by the judge, Jed S. Rakoff, came in response to a routine procedural motion by Times lawyers to rule in its favor, which defendants have a right to do after the plaintiff has presented all of its evidence to the jury. He found those claims by The Times convincing but also criticized the newspaper’s error as an example of “very unfortunate editorializing.”

His ruling also set up an awkward dynamic, coming as jurors were deliberating just down the hall from the courtroom. Though the judge instructed them not to read any media coverage of the trial, some legal experts criticized him for making public a decision that could have influenced their verdict if they had learned of it.

There were no immediate signs that had happened. The jury deliberated for roughly five hours on Tuesday before announcing its verdict just after 2:30 p.m.