Mr. Cohen will not be allowed to publicly discuss matters related to the continuing special counsel investigation into the Trump campaign’s ties to Russian election manipulation efforts. Those topics were sure to have been covered on Tuesday in his closed-door appearance with the Senate Intelligence Committee and will be again on Thursday in secret testimony to the House Intelligence Committee.

Mr. Cohen’s testimony — coming as the special counsel, Robert S. Mueller III, prepares to conclude his investigation, and with House Democrats starting their own — signals the beginning of what could be a perilous new phase for Mr. Trump’s presidency.

Mr. Gaetz’s threat and the enraged reaction to it reflected the stakes on Capitol Hill in anticipation of Wednesday’s open hearing.

Neither Mr. Cohen nor members of the Senate Intelligence Committee provided details of his testimony on Tuesday. Mr. Cohen began his meeting with the senators by apologizing for lying to them in 2017 about the duration of time during the 2016 campaign that the Trump Organization was in discussions about a Trump Tower project in Moscow, people familiar with what took place said.

Mr. Cohen emerged from a secure room in the Senate after more than eight hours of questioning and told reporters that he appreciated “the opportunity that was given to me” to tell the truth.

“I look forward to tomorrow to be able to, in my voice, tell the American people my story,” he said. “I am going to let the American people decide exactly who’s telling the truth.”

Senators who sat through the session indicated that Mr. Cohen’s testimony had made a strong impression on them. Senator Susan Collins, Republican of Maine, said Mr. Cohen appeared to be “humbled by the experience he has been through and what he knows lies ahead of him.”