Democrats were clearly nervous about what might happen during the White House review period. Representative Adam B. Schiff of California, the Intelligence Committee’s top Democrat, said the Democrats had already shared their memo with the F.B.I. and the Justice Department and would seek specific explanations from the department and the White House for any changes they might request.
“I think it’s going to be very hard for the White House, like it was hard for the Republicans on the committee, to block the release of this,” Mr. Schiff told reporters after the vote. “I am more concerned that they would make political redactions.”
A White House official said on Monday that it was prepared to review the memo.
“We will consider it along the same terms that we considered the Nunes memo — which is to allow for a legal review — national security review — led by the White House Counsel’s Office,” a White House spokesman, Raj Shah, told reporters aboard Air Force One.
But the memo’s fate is uncertain. Mr. Trump signaled earlier on Monday that he had little good will toward the committee’s Democrats, launching a broadside at Mr. Schiff. Mr. Trump accused Mr. Schiff on Twitter of illegally leaking confidential information from the committee, called the congressman “Little Adam Schiff” and ominously said that he “must be stopped.”
In a separate tweet later in the morning, Mr. Trump praised Representative Devin Nunes of California, who spearheaded the Republican memo as the committee chairman, calling him a “Great American Hero for what he has exposed and what he has had to endure.”
Automated Twitter accounts, called bots, then appeared to push the “Little Adam Schiff” hashtag on the social media platform.
Republicans framed the committee’s vote on Monday night as a matter of transparency.
“Obviously, the Democrats want to tell a side of the story that was different from the side we told,” said Representative K. Michael Conaway of Texas, a senior Republican on the committee. “And I think that’s fair to let them have their say in the public arena.”
Mr. Nunes’s three-and-a-half-page memo centered on the F.B.I.’s use of material from a former British spy, Christopher Steele, in the warrant application to spy on Mr. Page. Mr. Steele was researching possible connections between Russia’s election interference and Trump associates, but the memo said that the F.B.I. had not disclosed to a Foreign Intelligence Surveillance Court judge that he was being paid by the Democratic National Committee and lawyers for Hillary Clinton’s presidential campaign.
Despite Mr. Trump’s claims, the Republican memo did nothing to clear him of either collaborating with the Russians or obstructing justice — the two lines of inquiry being pursued by the special counsel, Robert S. Mueller III. The memo undermined an effort by some Republicans to cast doubt on the roots of the investigation by confirming that contacts between another former Trump foreign policy adviser, George Papadopoulos, and Russian intermediaries — not the material from Mr. Steele — were the primary factor in the opening of the investigation in July 2016.
Nevertheless, Democrats have denounced the document as a tactic to undermine the investigation and to protect Mr. Trump, and they have said it is riddled with errors and omissions.
Specifically, the Democratic memo is said to contend that the F.B.I. was more forthcoming with the surveillance court than Republicans had claimed. People familiar with the Democratic document said that it reveals that while the F.B.I. did not name the Democratic National Committee or Mrs. Clinton’s campaign, the bureau disclosed to the court that the information it had received from Mr. Steele was politically motivated.
Confronted on Monday with Democratic claims that such a disclosure was included in the application, Mr. Nunes conceded on “Fox & Friends” that there had been a “footnote” to that effect. But he called its inclusion a “far cry from letting the American people know that the Democrats and the Hillary campaign paid for dirt that the F.B.I. then used to get a warrant.”
Democrats also contest Republican claims that Andrew G. McCabe, the deputy director of the F.B.I. at the time, had testified before the Intelligence Committee late last year that the agency would not have sought a wiretap of Mr. Page without Mr. Steele’s dossier of information.
The F.B.I. suspected that Mr. Page, a former Moscow-based investment banker who was under investigation once before, was acting as a Russian agent.
The New York Times filed a motion on Monday asking the Foreign Intelligence Surveillance Court to unseal all materials related to the wiretap of Mr. Page, including the F.B.I.’s application for the warrant and other court documents. Since Congress enacted the Foreign Intelligence Surveillance Act in 1978, no such wiretapping application materials have been made public.
Monday’s vote was an about-face for the Intelligence Committee. Republicans on the committee voted against releasing the Democratic rebuttal last Monday during the same meeting they chose to initiate the release of their own. They argued then that the memo should first be shared with all members of the House — as the Republican memo had been — before being reconsidered for public release.
In initiating the release of the Democratic memo, the committee is relying on the same obscure House rule it invoked last week. The rule allows the Intelligence Committee to sidestep the usual back-and-forth between lawmakers and the executive branch over the government’s most closely held secrets if the committee deems release to be in the public interest.
At least some Republicans, meanwhile, signaled Monday that they had already begun to move on.
Mr. Nunes said he was pushing forward in secret with what he referred to on Friday as “Phase Two” of the majority’s investigation. He has said he is focused on the State Department under President Barack Obama and its role in the early stages of the Russia investigation, but he offered few additional details.