Ms Trump said in a rare public statement on Saturday that the words her husband used in the 2005 footage released on Friday "are unacceptable and offensive to me."
She added that the words do "not represent the man that I know," adding, "He has the heart and mind of a leader."
Trump was newly married when he bragged on tape about trying to have sex with married women and groping others without permission.
Ms Trump says she "hopes people will accept his apology, as I have, and focus on the important issues facing our nation and the world."
Trump vows to stay in race after calls for him to quit over lewd remarks
US presidential nominee Donald Trump vowed on Saturday to remain in the race even as his campaign was thrown into crisis as both his wife and running mate criticized him and more than a dozen prominent Republicans withdrew support and urged him to drop out following news of a recording of him making lewd comments in 2005.
Trump told the Wall Street Journal on Saturday morning "zero chance I'll quit." He also told The Washington Post he may deliver a speech Saturday afternoon to address concern among supporters and reiterate his determination to stay in the race.
The video was the latest calamity for Trump, who had hoped to revive his flagging campaign in the face of a recent drop in polls with less than a month until Election Day. The 2005 video of Trump talking on an open microphone showed the then-reality TV star speaking openly about groping women and trying to seduce a married woman. The video was taped only months after Trump married his third wife, Melania.
In a statement, Melania Trump called her husband's words "unacceptable and offensive to me."
"This does not represent the man that I know," Melania Trump said. "He has the heart and mind of a leader. I hope people will accept his apology, as I have, and focus on the important issues facing our nation and the world."
The backlash against the video was swift and widespread.
In an unusual move, vice presidential running mate Mike Pence issued a critical statement, saying on Twitter that he cannot defend the nominee.
"As a husband and father, I was offended by the words and actions described by Donald Trump in the eleven-year-old video released yesterday," Pence, who is governor of Indiana, said in a statement. "I do not condone his remarks and cannot defend them."
Pence indicated he would continue to support Trump, despite calls from several Republicans that the New York real estate mogul step aside and let Pence be the nominee.
There is no precedent for a major party to replace its nominee this late in the campaign and it is unclear if there is an avenue to force him out. Voting has begun in several states, including the important swing states of Virginia and North Carolina.
A hastily recorded apology by Trump early Saturday did not stymie an avalanche of calls from members of his party to quit.
Trump huddled on Saturday afternoon in Trump Tower with senior advisors, including New Jersey Governor Chris Christie and former New York City Mayor Rudy Giuliani.
Trump quickly moved to conduct damage control in Saturday's video in which he declared himself a changed man and attempted to shift the focus to his opponent, Democrat Hillary Clinton. He threatened, again, to focus his attacks on the infidelities of her husband, former President Bill Clinton, saying he would talk more about the pasts of both Clintons with only a month until the Nov. 8 election. Trump has dismissed questions about his own marital infidelities as irrelevant.
"Anyone who knows me knows these words don't reflect who I am. I said it, I was wrong, and I apologize," Trump said in his video statement, posted on his Facebook page.
The video overshadowed the publication of excerpts of Clinton's paid closed-door speeches made public on Friday by a hacker who claimed to have obtained them from the email account of John Podesta, the chairman of the Democrat's campaign.
In the speeches, she advocates for more open borders and trade, a position she abandoned during the primary because it was untenable to Democratic progressives. Trump has repeatedly criticized her for her past support of free trade.
Comments condemned
Trump has struggled to win over women voters, and the video is expected to further feed Democratic criticism about his past behavior toward women.
"I did try and fuck her. She was married," Trump said about one woman, before discussing his attraction to others.
"I just start kissing them," he said. "And when you're a star they let you do it."
"Grab them by the pussy. You can do anything," Trump said.
House Speaker Paul Ryan disinvited Trump to a scheduled appearance on Saturday afternoon in Wisconsin. Pence declined to speak in his place.
The list of Republicans announcing they would not vote for Trump or calling on him to step aside grew on Saturday: Senators Kelly Ayotte, Mark Kirk, Jeff Flake, John Thune, Mike Crapo,
Shelley Moore Capito and Mike Lee; House members Jason Chaffetz, Mia Love, Joe Heck, Bradly Byrne, Martha Roby and Barbara Comstock; and Governors John Kasich, Dennis Daugaard and Gary Herbert. Additionally, former presidential candidates Mitt Romney and Carly Fiorina also called on Trump to quit.
Capito, of West Virginia, called his remarks "disgusting and demeaning."
Chaffetz of Utah, one of Clinton's fiercest critics, retracted his endorsement of Trump, telling CNN he would not be able to look his 15-year-old daughter in the eye if he voted for him.
Other prominent Republicans, however, indicated they would stick with Trump. Ralph Reed, head of the Faith and Freedom Coalition, and Tony Perkins, head of the conservative Family Research Council, said they would continue to support him.
Conservatives point to the fact that the winner of the November election will get to appoint a Supreme Court justice as reason to stick with Trump despite the video. Greg Mueller, a conservative Republican strategist, pointed to Clinton's views on abortion as a reason religious voters will stick with Trump.
"Nothing indefensible that Donald Trump said 20 years ago is going to change that," he said. "Plus, to many religious voters, Mrs. Clinton is the epitome of a corrupt politician."
Representative Jack Kingston, a Republican from Georgia, argued Trump has changed since the video was filmed.
"I think 10 years ago he was a different man ..., I am very glad that he quickly apologized," Kingston said.
Evangelical leaders stick with Trump
Leaders of US religious conservative groups are largely standing behind Donald Trump even after vulgar sexual comments he made about women surfaced.
Most evangelical leaders did not condemn the Republican candidate and instead pointed to an urgent need to prevent Democratic nominee Hillary Clinton from winning the presidency, reshaping the Supreme Court and implementing liberal policies.
The latest blow to Trump's campaign came after a 2005 video surfaced of the then-reality TV star talking about groping women and trying to seduce a married woman. Vice presidential running mate Mike Pence said he could not defend Trump's words.
Gary Bauer, chairman of the Campaign for Working Families, said Trump's "grossly inappropriate language" does not change the choice facing the country in the November 8 election and that "I continue to support the Trump-Pence ticket".
"Hillary Clinton is committed to enacting policies that will erode religious liberty, promote abortion, make our country less safe, and leave our borders unprotected," Bauer said.
White evangelicals make up about 20 per cent of the US population, according to the Public Religion Research Institute, and represent a crucial voting bloc Trump needs to win the presidency.
They have long represented a pillar of support for Republicans. In 2004, they were instrumental in President George W Bush's re-election. They turned out in similar numbers in 2008 and 2012, when Mitt Romney, a Mormon who many evangelicals considered too moderate, was the Republican nominee, according to a report by the Pew Research Center.
Support from evangelicals for Trump has been strong throughout his campaign, even though it was only late in life that the New York businessman adopted their cause. Social conservatives flocked to his side over other deeply religious Republican presidential candidates, such as Ted Cruz.
"Naturally I'm disappointed," said Steve Scheffler, head of the Iowa Faith and Freedom Coalition. "But, you know, the Bible tells me that we are all sinners saved by grace and I don't think there's probably a person alive that I know of that hasn't made some mistakes in the past."
"So yes, I will vote for Donald Trump. I'm not excusing his behaviour at all. It's disgusting," he said.
Still, politically active Christian conservative leaders across the country said they were worried that Trump's comments could depress turnout among evangelicals.
"Evangelicals are not going to vote for Hillary," said religious political activist David Lane. "But this could cause them to stay home. This could be a big deal. Things like this matter."
Much will hinge on Trump's performance in the second presidential debate on Sunday night, and whether he can convince Christians that he is a changed man, Lane said.
Other religious leaders, however, were less forgiving.
Russell Moore, president of the Ethics & Religious Liberty Commission of the Southern Baptist Convention, tweeted an article detailing evangelical apathy toward the Trump tape, calling it a "disgrace."
"What a scandal to the gospel of Jesus Christ and to the integrity of our witness," Moore wrote.
Still, the majority view among religious conservatives appeared to be summed up by Tony Perkins, president of the Family Research Council action group, who said evangelicals "are left with a choice of voting for the one who will do the least damage to our freedoms."
