But in words broadcast live to millions of viewers, Mr. Trump, without evidence, alleged that former Vice President Joseph R. Biden Jr. and his son had profited from dealings in Ukraine. He insulted journalists and accused The Washington Post of publishing a “fake article.” He asserted without evidence that Senator Chris Murphy of Connecticut had “threatened” Ukrainian officials and claimed that Democrats had timed their impeachment inquiry to disrupt his trip to the United Nations.

Cutting away from the remarks would deprive Mr. Trump of a pulpit for his claims. But news producers often defer to the public’s right to know what its president is saying, especially at a crucial time in the country’s politics.

Several news networks on Wednesday chose to fact-check Mr. Trump after the fact.

“The president leveled several distortions, falsehoods in the course of that 45 minutes,” the CNN anchor Wolf Blitzer told viewers once Mr. Trump concluded his remarks. Mr. Blitzer’s network colleague Jeffrey Toobin called the appearance a “torrent of lies.”

On ABC, the anchor George Stephanopoulos appeared onscreen halfway through the event, informing viewers that there was “no evidence” to support Mr. Trump’s claims about the Bidens. Andrea Mitchell, on NBC, said the accusations against Mr. Murphy were “contrary to any information we have.”

“We don’t know what he’s talking about,” Ms. Mitchell added.

With the 2020 campaign looming, television producers are likely to face more of these thorny choices. “Broadcast journalists in particular are reassessing how they deal with this president in a live context,” said Mark Lukasiewicz, a former NBC News producer and now the dean of Hofstra University’s school of communication.