A federal judge on Wednesday put on hold a proposed plea deal between Hunter Biden and the Justice Department that would have settled tax and gun charges against the president’s son, stunning the courtroom and raising legal and constitutional questions about the agreement.

After moments of high drama in which the deal appeared headed toward collapse, the judge, Maryellen Noreika of the Federal District Court in Wilmington, Del., sent the two sides back to try to work out modifications that would address her concerns and salvage the basic contours of the agreement.

Under the proposed deal, Mr. Biden would have pleaded guilty to two tax misdemeanors and averted prosecution on a gun charge by enrolling in a two-year diversion program for nonviolent offenders.

Prosecutors and Mr. Biden’s team had both started the day confident that the proceeding would go smoothly and the judge would sign off on the deal immediately. As he entered the courtroom, Mr. Biden drew a deep breath and plunged forward to greet the prosecutors who investigated him for five years with handshakes and a smile.

But Judge Noreika had other ideas, telling the two sides repeatedly that she had no intention of being “a rubber stamp,” and spending three hours sharply questioning them over nearly every detail of the deal.

“I cannot accept the plea agreement today,” said Judge Noreika, who was nominated to the bench by former President Donald J. Trump in 2017 with the support of Delaware’s two Democratic senators.

An exhausted-looking Mr. Biden trudged out of the courthouse looking a bit stunned, as his lawyers puzzled over what to do next. At the end of the hearing, Mr. Biden entered a plea of not guilty on the tax charges, which he will reverse if the two sides revise their agreement to the judge’s satisfaction.

The muddled outcome only underscored how Mr. Biden’s personal and legal troubles have become an entrenched political issue in Washington, where Republicans have long sought to show that his foreign business ventures were aided by, or benefited, President Biden.

Those efforts have only intensified as Mr. Trump’s legal troubles have deepened and Republicans in Congress have sought to undercut the president heading into the 2024 election.

Republicans have accused David C. Weiss, the Trump-appointed U.S. attorney in Delaware who was retained by the Biden administration to complete the investigation into Mr. Biden, of cutting a “sweetheart deal” intended to help Democrats.

They have sought to cast the Biden family as corrupt and assailed the proposed deal as far too lenient, citing testimony from two I.R.S. investigators as evidence that the Justice Department had hamstrung the investigation and that President Biden played a role in his son’s business deals with companies and partners in Ukraine and China.

Hunter Biden’s foreign business ventures raised ethical concerns, especially while his father was vice president, and his personal problems — he has acknowledged being admitted to crack cocaine for a number of years — have given conservatives an endless stream of material to assail him. But Republicans have produced no compelling evidence that President Biden used his office to help his son in any substantive way.

The White House declined to comment directly on Wednesday’s court proceeding while communicating the president’s support for his son’s efforts to put his problems behind him.

“Hunter Biden is a private citizen, and this was a personal matter,” Karine Jean-Pierre, the White House press secretary, told reporters on Wednesday. “As we have said, the president, the first lady, they love their son, and they support him as he continues to rebuild his life.”

Judge Noreika’s concerns appeared to center on two elements of the proposed deal: a provision that would have offered Mr. Biden broad insulation against further prosecution on matters scrutinized by federal prosecutors during the five-year inquiry, and the diversion program on the gun charge, under which she would be called on to play a role in determining whether Mr. Biden was meeting the terms of the deal.

Judge Noreika said she was not trying to sink the agreement, but to strengthen it by ironing out ambiguities and inconsistencies, a view held by some former department officials.

“The judge appropriately wanted to make sure that the parties were clear on whether Hunter Biden could be prosecuted for additional crimes in the future,” said Barbara L. McQuade, who was the U.S. attorney for the Eastern District of Michigan from 2010 to 2017.

Judge Noreika kicked off the hearing by telling lawyers that they did not need to keep “popping” up and down every time she asked them a question.

It was a signal that she was about to subject them to a relentless interrogation over elements of an agreement she described, variously, as “not standard, not what I normally see,” possibly “unconstitutional,” without legal precedent and potentially “not worth the paper it is printed on.”

Judge Noreika quickly zeroed in on a paragraph offering Mr. Biden broad immunity from prosecution, in perpetuity, for a range of matters scrutinized by the Justice Department. The judge questioned why prosecutors had written it in a way that gave her no legal authority to reject it.

Then, in 10 minutes of incisive questioning, she exposed serious differences between the two sides on what, exactly, that paragraph meant.

Christopher Clark, Mr. Biden’s lead lawyer, said it indemnified his client not merely for the tax and gun offenses uncovered during the inquiry, but for other possible offenses stemming from his lucrative consulting deals with companies in Ukraine, China and Romania.

Prosecutors had a far narrower definition. They saw Mr. Biden’s immunity as limited to offenses uncovered during their investigation of his tax returns dating back to 2014, and his illegal purchase of a firearm in 2018, when he was a heavy drug user, they said.

When the judge asked Leo Wise, a lead prosecutor in the case, if the investigation of Mr. Biden was continuing, he answered, “Yes.”

When she asked him, hypothetically, if the deal would preclude an investigation into possible violation of laws regulating foreign lobbying by Mr. Biden connected with his consulting and legal work, he replied, “No.”

Mr. Biden then told the judge he could not agree to any deal that did not offer him broad immunity, and Mr. Clark popped up angrily to declare the deal “null and void.”

The disagreement over such a central element of the deal was remarkable, given the months of negotiations that went into reaching it.

“Today was very unusual, but based on my experience, I think the deal will now get done,” said John P. Fishwick Jr., who served as U.S. attorney for the Western District of Virginia from 2015 to 2017. “Judges are reluctant to reject deals but do ask questions. These should have been cleared up before today’s hearing, but they were not, so she helped provide more clarity.”

The 30 journalists in the gallery then witnessed a remarkable tableau of real-time, public deal-making. With the judge having called a recess, the defense and prosecution teams first separated into two packs, then merged into a circle to hash out a new compromise. An unsmiling Mr. Weiss paced back and forth, jaw tense and hands jammed into the pockets of his suit.

After an official recess was declared, Mr. Clark agreed to the narrower terms on Mr. Biden’s behalf.

But Judge Noreika still appeared to be unconvinced. She turned her attention to the fine print of the deal that had been struck on the gun offense, requiring Mr. Biden to avoid using drugs or owning a firearms during the two-year diversion program.

She objected strenuously to how a violation of its terms would be handled.

Typically, the Justice Department could independently verify any breach and bring charges. But Mr. Biden’s team, concerned that the department might abuse that authority if Mr. Trump is re-elected, successfully pushed to give that power to Judge Noreika, arguing that she would be a more neutral arbiter.

Judge Noreika suggested that such an arrangement could be unconstitutional because it might give her prosecutorial powers, which were vested in the executive branch by the Constitution.

“I’m not doing something that gets me outside my lane of my branch of government,” said the judge, adding, “Go back and work on that.”

Erica L. Green contributed reporting.