The federal judge overseeing former President Donald J. Trump’s prosecution on charges of seeking to overturn the 2020 election rejected his request on Friday to be able to speak broadly about evidence and witnesses — and warned Mr. Trump she would take necessary “measures” to keep him from intimidating witnesses or tainting potential jurors.

The caution from the judge, Tanya S. Chutkan, came during a 90-minute hearing in Federal District Court in Washington to discuss the scope of a protective order over the discovery evidence in Mr. Trump’s case, a typically routine step in criminal matters. Judge Chutkan said she planned to impose the order but agreed to a modification requested by the Trump legal team that it apply only to “sensitive” materials and not all evidence turned over to the defense.

She concluded the hearing with a blunt warning to Mr. Trump, and an unmistakable reference to a recent social media post in which he warned, “If you go after me, I’m coming after you!” — a statement his spokesman later said was aimed at political opponents and not at people involved in the case.

“I do want to issue a general word of caution — I intend to ensure the orderly administration of justice in this case as I would in any other case, and even arguably ambiguous statements by the parties or their counsel,” she said, could be considered an attempt to “intimidate witnesses or prejudice potential jurors,” triggering the court to take action.

“I caution you and your client to take special care in your public statements in this case,” she added. “I will take whatever measures are necessary to protect the integrity of these proceedings.”

The hearing was the first major legal skirmish between prosecutors working for the special counsel, Jack Smith, and the Trump legal team and involved a clash over the core issue that makes the case different than any other: Mr. Trump’s intention to make the criminal case the center of his presidential campaign and to publicly criticize some witnesses, including a political opponent, former Vice President Mike Pence.

Judge Chutkan, in her first appearance as trial judge in the case, made it clear — within minutes of ascending the bench — that she intended to view Mr. Trump primarily as a defendant rather than a political figure, and suggested she sided with the government’s push for a speedy trial.

“Mr. Trump, like any other American,” has a right to free speech under the First Amendment, she said.

But that right, she added, “is not absolute.”

The judge dismissed the argument made by John F. Lauro, Mr. Trump’s defense lawyer, who complained that the protective order would put “a chill” over his client’s ability to defend himself in the 2024 campaign.

“He is a criminal defendant,” Judge Chutkan said. “He is going to have restrictions like every other criminal defendant” — adding that she was not going to “allow him any greater or lesser latitude than any defendant in a criminal case.”

The judge described Mr. Trump’s candidacy as “a day job,” like another defendant.

The hearing marked the first time that Mr. Trump’s legal team and Justice Department prosecutors appeared in front of Judge Chutkan — an Obama appointee Mr. Trump and his allies began attacking soon after she was selected.

Judge Chutkan repeatedly made it clear that she intended to hold Mr. Trump accountable for statements he made outside the courtroom that could potentially endanger witnesses or others involved in the case, reminding Mr. Lauro that the terms of Mr. Trump’s release precluded witness tampering and intimidation.

Witnesses, she said, “may not have the type of protections that he has.” She added, “I see the possibility for a lot of problems here.”

Thomas P. Windom, a lead prosecutor in the case, said Mr. Trump and his legal team have already begun “to try the case in the media” and intended to use evidence “for political purposes.” He cited Mr. Lauro’s recent appearances on Sunday news shows this week.

Mr. Lauro, while saying Mr. Trump would abide by the court’s ruling, said that virtually any restraints on what Mr. Trump might say in public would impede his ability to conduct his campaign, and specifically raised the problem posed by Mr. Pence — who has spoken extensively about Mr. Trump’s actions on the stump.

These are “uncharted waters,” he said. “President Trump has the right to respond.”

The hearing came a day after Mr. Smith proposed a rapid schedule for the case, with a trial starting on Jan. 2, 2024. Mr. Trump has argued that holding a trial during an election year is tantamount to political interference, and his lawyers have sought to delay the proceedings in Washington, as they have in the Florida documents case.

Mr. Trump, his lawyers and allies have cast both the substance of the four-count indictment handed down Aug. 1, and the government’s efforts to limit his statements about the case outside the courtroom, as an attack on the former president’s First Amendment rights.

The order Judge Chutkan was set to put in place was not a gag order, but a more routine measure designed to restrict the public release by Mr. Trump of any of the underlying discovery evidence collected by the government during more than two and a half years of investigation.

In many criminal cases, courts issue such so-called protective orders to keep participants from releasing details that might taint the pool of prospective jurors. But the stakes in this case, coupled with Mr. Trump’s history of incendiary public statements, are far higher.

Prosecutors are legally bound to give that evidence to Mr. Trump’s lawyers. They often ask a judge to bar the lawyers or their clients from using the evidence outside of normal channels, like pretrial motions, so that the case can proceed with a measure of order and decorum.

But the legal spat over the order became ensnared in arguments about free speech when prosecutors last week brought up the issue of Mr. Trump’s habit of making menacing social media posts. They drew the judge’s attention specifically to a vague but threatening message that Mr. Trump had written just one day after he was arraigned in the case.

That gave Mr. Trump’s lawyers the opportunity to swing the case back to their chosen defense theme — the First Amendment. The lawyers responded that prosecutors were seeking to make Judge Chutkan “assume the role of censor” and to squelch Mr. Trump’s ability to communicate with the public about a high-profile criminal matter.

The prosecutors argued in favor of moving ahead in an orderly fashion and avoiding the typical media frenzy. They warned in court papers before the hearing that if Mr. Trump were allowed to speak openly about the evidence in his case outside of court it could threaten witnesses, taint the jury pool or otherwise harm “the integrity” of the proceeding.

The former president’s lawyers framed the prosecution — and the protective order — as the government’s attempt to silence Mr. Trump.

The imposition of the protective order was a small but important step in the nuts-and-bolts process of prosecuting Mr. Trump. It meant that the government could start turning over discovery evidence to Mr. Trump’s lawyers — a move that will give them a sense of the government’s case against their client and allow them to start mapping out pretrial motions or even trial defenses.

The “sensitive materials” Judge Chutkan was seeking to protect were things like “personally identifying information” concerning witnesses in the case and any information that had emerged from the grand jury that investigated the former president. (Grand juries work under strict secrecy rules.)

Mr. Trump’s legal team could show him the sensitive materials, but under the proposed order they would not be allowed to give him copies. He would also not be permitted to write down any personal information about people mentioned in the materials.

Moreover, while the sensitive discovery evidence could be used to file motions in the case, those motions would have to be partly redacted or submitted under seal.