Other Republicans cast the author of the op-ed as a renegade whose anonymous status undercut any credibility.
In one eye-opening passage, the writer says cabinet members initially considered invoking the 25th Amendment of the US Constitution, which provides for the president's removal if he is "unable to discharge the powers and duties of his office".
The writer of the piece says he or she wants Americans to know that "there are adults in the room".
The unsigned column drew disavowals from at least 11 advisers to Trump, including Vice President Mike Pence, Ambassador to Russia Jon Huntsman, Secretary of State Mike Pompeo and Defense Secretary James Mattis.
"Virtually you know it's treason", Trump told Fox News' Pete Hegseth.
The writer describes the president as "impetuous, adversarial, petty and ineffective".
Dana Perino, the former White House press secretary under George W. Bush, called the mysterious author of the op-ed "extremely self-indulgent".
Yet senior GOP lawmakers rejected the idea that the op-ed would damage Trump, insisting instead that it would help him during the midterms and in the 2020 presidential election.
Veteran journalist Carl Bernstein on Thursday called for Congress to hold hearings on President TrumpDonald John TrumpNorth Korea to cooperate with United States in nuclear talks: report Nike to air Kaepernick ad during 'Thursday Night Football: report Brothel owner who won Nevada GOP primary being investigated over sexual assault allegations MORE's fitness for office.
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The Los Angeles County District Attorney's office said the alleged incident was outside California's statute of limitations. Neither Kevin Spacey nor Steven Seagal will face charges for alleged sexual assaults that took place about 25 years ago.
Vice President Mike Pence also denied writing the opinion piece.
The Times op-ed, and Woodward's book, which is to be published next week, followed many news articles during Trump's 19-month presidency that have depicted turbulence inside the White House under his leadership.
Meanwhile, First Lady Melania Trump said: "If a person is bold enough to accuse people of negative actions, they have a responsibility to publicly stand by their words". She praised the free press as "important to our democracy" but assailed the writer, saying, "You are not protecting this country, you are sabotaging it with your cowardly actions". Democrats continue to call for an adjournment over the handling of documents from Kavanaugh's tenure in the George W. Bush White House.
"Our office is above such amateur acts".
With many prominent administration members delivering on-the-record denials, the focus could now fall on other senior aides to do the same, with questions raised about those who stay silent.
Bernstein said that Trump's behavior is "no surprise" to Republicans in Congress.
"Nothing in this town stays secret forever, and so ultimately I do think we will find out who is the author", he said.
"I don't really know what's going on in the White House. but it seems to me from outside it's open-ended down there, that there's no confidentiality and not a lot of loyalty down there with a lot of people", he said.
As Yashar Ali of New York Magazine noted, Trump officials have been known to purposely copy language of other senior staffers when leaking news in an effort to throw the paranoid president off their trail.