Schwarzenegger's remarks, as he kicked off a four-day bus tour of the state in the final days of the recall campaign, came after a Los Angeles Times story published Thursday in which six women accused him of sexually harassing and groping them.
"When I am governor I want to prove to the women that I will be a champion for the women, a champion of the women," Schwarzenegger said at a rally in San Diego. "Now let's go from the dirty politics to the future of California."
The crowd of Schwarzenegger supporters interrupted him with cheers.
Schwarzenegger said: "Yes, I have behaved badly some times, yes it is true that I was on rowdy movie sets and I have done things that were not right which I thought then was playful but now I recognize that I offended people."
"Those people that I have offended, I want to say to them I am deeply sorry about that and I apologize because that's not what I'm trying to do," he said.
The charges were overshadowing Schwarzenegger's four-day barnstorming tour through the state, which began in San Diego Thursday morning.
Schwarzenegger spokesman Sean Walsh had denied the allegations in comments to the Times.
"We believe Democrats and others are using this to try to hurt Arnold Schwarzenegger's campaign," Walsh said. "We believe that this is coming so close before the election, something that discourages good, hard-working, decent people from running for office."
The charges were not a total surprise, reports CBS News Correspondent Steve Futterman: Schwarzenegger aides have been prepared for this type of article for some time. Throughout the campaign they have been asked about allegations and whispers concerning Schwarzenegger's behavior towards women, and Schwarzenegger himself has said he expected negative stories about his personal life.
Schwarzenegger's alleged past indiscretions have been an issue in the campaign since he announced his bid for governor. Much of the controversy has surrounded a 1977 interview in Oui magazine in which Schwarzenegger recalled engaging in group sex at Gold's Gym. The actor has said he doesn't remember the incident.
The new allegations came as momentum has been building for the Schwarzenegger camp, with a new poll showing the actor with a big lead over all other candidates.
Gov. Gray Davis, who is in jeopardy of becoming only the second governor in the nation to be recalled, characterized the recall as a two-man race between him and Schwarzenegger that he said provided a "shockingly clear choice."
Schwarzenegger's bus tour plans stops and rallies in Southern California, the Central Valley and the San Francisco Bay area, building up to a march on the Capitol in Sacramento on Sunday.
The bus tour will heavily advertise the candidate's Hollywood ties. Schwarzenegger will be riding on a bus nicknamed Running Man, the campaign staff on Total Recall, and the press on buses nicknamed Predator 1 through 3 and True Lies.
In Sacramento Wednesday, Schwarzenegger offered a blueprint for his first 100 days in office, repeating promises to repeal the tripling of the state car tax and to have the budget audited.
"We are ready to take office," he told a crowd of about 400 supporters. "We are ready to take action. We are ready to return California to the people."
Davis' campaign spokesman, Peter Ragone, countered that it was a mistake for Schwarzenegger to "start measuring the drapes in the offices of the Capitol."
Davis appeared Wednesday with retired Gen. Wesley Clark, a presidential candidate and the latest high-profile Democrat to visit California in support of the governor.
He later appeared with his newest convert, independent Arianna Huffington, who dropped out of the race Tuesday but promised to work with Davis to defeat the recall and keep Schwarzenegger out of office.
The Times said it interviewed the women over the last seven weeks during an investigation into whether Schwarzenegger harassed women. None of the women approached the newspaper on her own or has filed legal action against the actor.
Two of the women allowed their names to be used; four spoke on condition of anonymity.
E. Laine Stockton told the Times she was groped by Schwarzenegger in 1975 at Gold's Gym in Venice Beach. Then 19, Stockton claimed the bodybuilder came up from behind her and reached under her T-shirt to grab her breast, and then silently walked away.
"I was just shocked, shocked to the point where I almost didn't know how to react, because it was so out of the blue and so unexpected," she told the newspaper.
Anna Richardson, a British television host, said Schwarzenegger touched her breast when she was interviewing him in 2000 on a promotional tour for "The Sixth Day." Richardson, who previously told Premiere magazine about the alleged incident, was accused by a Hollywood publicist then working with Schwarzenegger of fabricating the story.
Others interviewed included two crew members working in 1990 on "Terminator 2: Judgment Day." One woman, now 41, said Schwarzenegger groped her at least three times in an elevator at the hotel where the cast and crew was staying.
The other woman, now in her 30s, said Schwarzenegger pulled her into his lap and whispered vulgarities while other men watched, smiling.
Three of the women who spoke on condition of anonymity said being named could jeopardize their careers; the fourth feared public ridicule or harm to her husband's business.
At Wednesday's campaign event, Schwarzenegger promised that he would repeal the tripling of the state car tax on his first day in office, then move to have the state budget audited and call a special session of the Legislature to deal with spending cuts.