Democrats have called the Republican document a dangerous effort to build a narrative to undercut the department’s investigation into whether Mr. Trump’s associates colluded with Russians and whether Mr. Trump obstructed justice. They say it uses cherry-picked facts assembled with little or no context and could do lasting damage to faith in federal law enforcement.

The F.B.I. statement ran counter to the decidedly low-key approach that Mr. Wray has taken as director, avoiding news media interviews and delivering anodyne speeches to law enforcement groups. He had worked quietly in the hopes of keeping the F.B.I. out of the president’s cross hairs.

Since taking over the F.B.I. about six months ago, Mr. Wray has had to defend the bureau against the president’s broadsides. But the director has done so in a nonconfrontational manner. In December, when Mr. Trump said the F.B.I.’s standing was the “worst in history” and its reputation in “tatters,” Mr. Wray sent a message to the bureau’s more than 35,000 agents and support staff saying that their professionalism and dedication were inspiring.

Stephanie Douglas, a former top F.B.I. executive, said Mr. Wray had to act on his concerns.

“His role as the F.B.I. director is about credibility,” she said. “He’s obligated by his role to speak the truth. I think he did the right thing. That’s his job. If he didn’t say something about a document lacking factual accuracy, he would have to make up for a lot of lost trust.”

Mr. Wray had strongly objected to the move to release the memo and was allowed to review it only on Sunday, after Mr. Nunes relented. Mr. Wray made a last-ditch effort on Monday, going to the White House with the deputy attorney general, Rod J. Rosenstein, to try to persuade the White House to stop the release of the memo. They spoke to John F. Kelly, the White House chief of staff, but were unsuccessful.

Mr. Rosenstein was also asked by the president last month whether he was “on my team,” according to an official briefed on the exchange. Mr. Rosenstein appeared surprised but responded affirmatively, according to CNN, which first reported the encounter.