Both Trump's wife and his running mate criticized his words, saying they were insulting and indefensible.
"The media and establishment want me out of the race so badly - I WILL NEVER DROP OUT OF THE RACE, WILL NEVER LET MY SUPPORTERS DOWN!" Trump wrote on Saturday afternoon on the social media website Twitter.
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.
Trump is due to appear alongside Democrat Hillary Clinton on Sunday in their second debate in the runup to the general election. Clinton is not expected to address Trump's video before then.
The 2005 video of Trump talking on an open microphone showed the then-reality TV star speaking 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," she 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 over the video was swift and widespread.
More than 60 prominent Republican current and former officeholders issued statements condemning Trump's remarks about women, including House Speaker Paul Ryan and John McCain, the party's standard bearer in 2008. More than 20 called for Trump to end his presidential bid.
In an unusual move, his vice presidential running mate Mike Pence issued a critical statement of Trump's words, saying on Twitter that he "cannot defend them."
"As a husband and father, I was offended by the words and actions described by Donald Trump," said Pence, who is governor of Indiana.
Pence indicated he would continue to support Trump, despite calls from several Republicans for Trump to 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 was unclear if there was an avenue to force him out. Voting has begun in several states, including swing states Virginia and North Carolina.
A recorded apology by Trump early on Saturday did not stymie an avalanche of calls from members of his party to quit.
Trump huddled on Saturday in Trump Tower with senior advisers, including New Jersey Governor Chris Christie and former New York City Mayor Rudy Giuliani.
Despite previous scheduling, Giuliani will appear on five major Sunday morning news programs, a rare round robin reserved for major news events - replacing Republican Chairman Reince Priebus on CBS's "Face the Nation" and campaign manager Kellyanne Conway on "Fox News Sunday," a last-minute switch. No reason was given for the bump of Conway. A CBS news release said the RNC asked to replace Priebus because Trump's operation wanted "a campaign person" to appear on the program.
Trump left the building briefly to greet a small crowd of supporters, saying "100 percent" he would remain in the race. Before returning to a bank of elevators, he told reporters, "Tremendous support."
He quickly moved to do damage control in Saturday's video in which he declared himself a changed man and attempted to shift the focus to his opponent Clinton and her husband, former President Bill Clinton. On Twitter, Trump posted critical statements from Juanita Broaddrick, a woman who has accused Bill Clinton of sexually assaulting her.
"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.
Trump has struggled to win over women voters, and the video was expected to further feed Democratic criticism about his past behavior toward women. Trump's support has suffered among suburban women and white, college-educated women, groups that Republicans have traditionally won.
In the recording that triggered the firestorm, Trump said of one woman, "I did try and fuck her. She was married." He went on to discuss 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.
On Saturday afternoon, CNN published a separate report detailing remarks Trump made over the course of several years while appearing on Howard Stern's radio program.
The remarks included discussing the size of his daughter's breasts and that he once had sex with three women at the same time. Trump was asked if he would have sex with a black woman and responded, "It depends on what your definition of black is."
The remarks were the last straw for some Republicans who have stuck with him through a series of controversial remarks, including calling Mexican immigrants "rapists" and "criminals," calling for a ban on Muslim immigrants, attacking a judge of Mexican descent, attacking the Gold Star family of a Muslim soldier killed at war and saying Senator John McCain was not a war hero because he had been a prisoner of war.
House Speaker Paul Ryan disinvited Trump to a scheduled appearance on Saturday 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, Lisa Murkowsi, Dan Sullivan, Mark Kirk, Jeff Flake, John Thune, Mike Crapo, Shelley Moore Capito and Mike Lee; House members Jason Chaffetz, Mia Love, Joe Heck, Bradley Byrne, Martha Roby and Barbara Comstock; and Governors John Kasich, Dennis Daugaard and Gary Herbert.
Additionally, former presidential candidates Mitt Romney and Carly Fiorina and former Secretary of State Condoleezza Rice called on Trump to quit.
"Donald Trump's behavior this week, concluding with the disclosure of his demeaning comments about women and his boasts about sexual assaults, make it impossible to continue to offer even conditional support for his candidacy," McCain said in a statement.
While Democrats largely remained silent, opting to let Republicans attack one of their own, Vice President Joe Biden wrote on Twitter, "The words are demeaning. Such behavior is an abuse of power. It's not lewd. It's sexual assault."
Some prominent Republicans 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.
"I think 10 years ago he was a different man," said Representative Jack Kingston, a Republican from Georgia. "I am very glad that he quickly apologized."