Prosecutors and lawyers for former President Donald J. Trump squared off on Monday in a federal appeals court in Washington to debate the validity of the gag order placed on Mr. Trump in the criminal case accusing him of plotting to overturn the 2020 election.

The hearing in front of the U.S. Court of Appeals for the District of Columbia Circuit followed more than a month of back-and-forth arguments about the order. It was put in place by the trial judge in October to stop Mr. Trump from maligning or threatening prosecutors, potential witnesses or court employees involved in the case.

From the start, the gag order has led to a momentous clash over how to protect people taking part in the election interference case from Mr. Trump’s barrage while preserving his rights as he campaigns for president and claims that the prosecution is political persecution.

When Judge Tanya S. Chutkan, an appointee of President Barack Obama, first imposed the order, she tried to thread that needle by barring Mr. Trump from lashing out at any people connected to the case — herself excepted — while still allowing him to say what he wants about what he asserts is the partisan and retaliatory nature of the case.

Mr. Trump’s lawyers appealed the order almost as soon as it was imposed, deriding it as “the essence of censorship.”

Each of the three judges on the appellate panel assigned to the case was nominated by a Democratic president: Judges Patricia Millett and Cornelia Pillard were both Obama appointees, and Judge Brad Garcia was appointed by President Biden.

In court papers, Mr. Trump’s lawyers have told the appeals court that the order should be repealed since it violates the First Amendment. They also said it represented an effort by Judge Chutkan to “micromanage” Mr. Trump’s “core political speech” before and during a trial that is scheduled to begin in March in the midst of the Republican primary season.

Prosecutors working for Jack Smith, the special counsel overseeing the federal prosecutions of Mr. Trump, have fired back that courts have wide discretion to limit the statements made by criminal defendants. They say that this gag order in particular was needed because of Mr. Trump’s “near daily” attacks against Mr. Smith, Judge Chutkan and potential witnesses in the case, including former Vice President Mike Pence and Gen. Mark A. Milley, the former chairman of the Joint Chiefs of Staff.

The prosecutors have tried to position themselves as protectors of both the integrity of the judicial process and the people who take part in it, telling the appeals court that Mr. Trump’s threats on social media have sometimes had damaging effects in the real world.

It is not clear how quickly the three-judge panel of the appeals court will decide on whether to rescind the gag order or keep it in place as the case moves toward its trial date. The gag order has been in abeyance for about two weeks as the court has gotten filings from the defense and the prosecution.

If the order is upheld and goes back into effect, Judge Chutkan may confront an even tougher issue: how to enforce the decree if Mr. Trump violates it. A violation of a gag order is treated as a matter of contempt of court, which could result in a reprimand, a fine or imprisonment. But how that would play out is complicated.

There are two types of contempt: civil, which is typically used to coerce future compliance with an order like making a recalcitrant witness testify; and criminal, which is focused on punishing past defiance of an order. Typically — though not always — judges have treated violations of gag orders as the latter type.

In federal court, judges cannot unilaterally impose a fine or order someone imprisoned for criminal contempt. Rather, such an accusation is treated as a new offense that requires the appointment of a prosecutor and another trial — including a right to a decision by a jury.

The battle over the federal gag order comes as a state appeals court in New York is considering the merits of two related gag orders imposed on Mr. Trump by Justice Arthur F. Engoron, who is overseeing his civil fraud trial in Manhattan.

Those orders — which are also currently paused — would bar Mr. Trump or any of his lawyers from targeting Justice Engoron’s law clerk. The clerk has suffered repeated attacks by the former president and his allies, who have accused her of being a Democratic partisan.

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