For much of this week, after a federal judge temporarily froze the gag order she imposed on him, former President Donald J. Trump has acted like a mischievous latchkey kid, making the most of his unsupervised stint.

At least three times in the past three days, he has attacked Jack Smith, the special counsel leading his federal prosecutions, as “deranged.” Twice, he has weighed in about testimony attributed to his former chief of staff Mark Meadows, who could be a witness in the federal case accusing him of conspiring to overturn the 2020 election.

Each of Mr. Trump’s comments appeared to violate the gag order put in place less than two weeks ago to limit his ability to intimidate witnesses in the case, assail prosecutors or otherwise disrupt the proceeding. And after the former president was fined $10,000 on Wednesday for flouting a similar directive imposed on him by the judge presiding over a civil trial he is facing in New York, federal prosecutors asked that he face consequences for his remarks about the election interference case as well.

On Friday, the judge who imposed the federal order, Tanya S. Chutkan, put it on hold for a week to allow the special counsel’s office and lawyers for Mr. Trump to file more papers about whether she should set it aside for an even longer period as an appeals court considers its merits.

But in the first round of those additional papers, prosecutors said on Wednesday that the order should be kept in place as the appeals court considers Mr. Trump’s request. They also said the lenient way in which Mr. Trump was released from custody after his indictment should be reconsidered for a simple reason: He has kept on violating the gag order’s provisions.

“The defendant has capitalized on the court’s administrative stay to, among other prejudicial conduct, send an unmistakable and threatening message to a foreseeable witness in this case,” wrote Molly Gaston, a prosecutor. “Unless the court lifts the administrative stay, the defendant will not stop his harmful and prejudicial attacks.”

In accusing Mr. Trump of persistently breaking the now-paused order, Ms. Gaston pointed to a social media message that the former president posted on Tuesday night, lashing out at Mr. Smith and dissecting statements attributed to Mr. Meadows in a news article.

Mr. Trump’s message also called the various people who have cooperated with the authorities in some of the prosecutions he is facing “cowards” and “weaklings.”

“I don’t think that Mark Meadows is one of them,” Mr. Trump wrote, “but who really knows?”

When Judge Chutkan initially imposed the order at a contentious hearing in Federal District Court in Washington, she said it was needed to keep Mr. Trump from targeting members of her staff, Mr. Smith or members of his staff, or anyone who might appear as a witness in the matter.

But with the order in abeyance, those are precisely the people Mr. Trump has gone after.

On Monday, before he issued his post about Mr. Meadows, Mr. Trump called Mr. Smith “deranged” again in a different post. Then on Wednesday, he repeated the attack on Mr. Smith and again brought up Mr. Meadows’s statements about the election case.

Those comments came while Mr. Trump was speaking to reporters in a courthouse in Manhattan where he is on trial in a separate proceeding: a civil case in which he stands accused of fraudulently inflating the value of his real estate holdings.

John F. Lauro, a lawyer representing Mr. Trump, appealed Judge Chutkan’s order shortly after she imposed it and a few days later asked her to suspend enforcing it pending the outcome of the appeal. In his motion for the stay, Mr. Lauro called the gag order “breathtakingly overbroad” and “unconstitutionally vague,” saying that it “violates virtually every fundamental principle of our First Amendment jurisprudence.”

Prosecutors fired back in their own motion on Wednesday, telling Judge Chutkan that she needed to lift her temporary stay to protect people involved in the case.

“Without the court’s intervention,” Ms. Gaston wrote, “the defendant will continue to threaten the integrity of these proceedings and put trial participants at risk.”

Shortly before prosecutors filed their papers, the American Civil Liberties Union released a proposed friend-of-the-court brief on its website siding with Mr. Trump and urging Judge Chutkan to re-evaluate her order.

In its brief, which has not yet been filed as part of the case, the A.C.L.U. said the gag order was both “impermissibly vague” and “impermissibly broad” and hindered Mr. Trump’s “ability to speak publicly about the substance of the prosecution.” The brief went on to note that Mr. Trump’s criminal case was “in many ways inextricable from the 2024 presidential campaign in which he is a declared candidate.”