Cain spoke at a televised news conference to respond to Sharon Bialek, a Chicago woman who went public on Monday with allegations that Cain made a crude sexual advance against her in 1997 when he was head of the National Restaurant Association.
Bialek is among at least four women to accuse Cain of sexual harassment. A woman who last week accused Cain of a series of unwanted advances spoke publicly for the first time and told The New York Times she had been warming to the idea of a "joint press conference where all of the women would be together" to lay out their cases.
Cain denied ever having seen Bialek until she appeared on television with her lawyer Gloria Allred to lay out allegations that he insisted "simply did not happen."
Bialek, he said, is a "troubled woman" produced by the "Democrat machine of America" to undermine his candidacy.
Asked if he is willing to submit to a lie-detector test to prove his innocence, the 65-year-old former pizza executive said: "Yes. I absolutely would. But I'm not going to do that unless I have a good reason to do that."
Cain is attempting to prevent his campaign from unraveling at a time when he has been leading in many polls of Republican voters considering who to choose as their presidential nominee to face President Barack Obama in the November 2012 election.
The drama is taking a toll. A Reuters-Ipsos poll on Tuesday found that 40 percent of Republicans view Cain less favorably after watching a video of Bialek's accusations.
Cain shrugged off the results of the poll, saying it is only natural to see a decline when the news media have covered the allegations against him so intensely.
ROMNEY WEIGHS IN
But some of Cain's rivals for the Republican nomination began to voice concerns, a day before the candidates gather in Michigan for a debate.
"These are serious allegations," Mitt Romney told an ABC News/Yahoo interview. "And they're going to have to be addressed seriously. I don't have any counsel for Herman Cain or for his campaign, they have to take their own counsel on this."
Another rival, Newt Gingrich, told a separate ABC/Yahoo interview: "Clearly Herman Cain has to answer the charges. He has to explain what happened. He has to do so in a way that's convincing, and I think that's unavoidable."
As Cain spoke, a woman who accused Cain last week and who had remained unidentified came forward and identified herself to The New York Times. Her name is Karen Kraushaar and she is a spokeswoman at the U.S. Treasury Department.
"When you are being sexually harassed in the workplace, you are extremely vulnerable," she told the Times. "You do whatever you can to quickly get yourself into a job some place safe, and that is what I thought I had achieved when I left."
ATTACKING ACCUSER'S CREDIBILITY
Cain was dismissive of Bialek, and his campaign has sought to attack her credibility by pointing out she had filed for bankruptcy in the past and been involved in civil lawsuits in Cook County, which includes Chicago.
An email from the Cain campaign questioned who was financing Bialek's legal team and "whether any media agreed to pay for her story."
Bialek said she had not sold her story while admitting she had declared bankruptcy after the death of her mother to help her father pay for medical bills and a custody battle.
"My whole objective is to tell the truth and also help other people out there who may have been in similar situations. And initially I went into this hoping every hope of hopes that Herman would just step forward. That was my actually primary goal," Bialek told CNN.
At his news conference, Cain said Bialek's accusations "exceed common sense, and they certainly exceed the standards of decency in America."
He spoke of his wife, Gloria, with whom he has been married for 43 years, saying that after watching Bialek's news conference she told him "that doesn't even sound like anything that you would ever do to anyone."
Gloria Cain shuns the political spotlight and has not spoken out about the harassment allegations, which became public more than a week ago.
