"Ain't gonna' happen," Cain said at a news conference a day after a fourth woman accused him of unwanted sexual advances.
"We will get through this," he added, trying to steady a campaign that has been rocked by the controversy for the past 10 days.
Cain denied anew that he had ever behavior inappropriately and said the alleged incidents "they simply didn't happen." He said he would be willing to take a lie detector test if he had a good reason.
Earlier in the day, Cain sought to undercut the credibility of the latest woman whose accusations are threatening his Republican presidential campaign. His chief rival, Mitt Romney, weighed in for the first time, calling the allegations "particularly disturbing."
Cain said he called the news conference because he wanted to speak directly to the public, accusing the media of distorting his response to the allegations. He said that had never seen Sharon Bialek until she called her news conference on Monday in New York, alongside attorney Gloria Allred.
"I don't even know who this woman is," he said of Bialek. "I tried to remember if I recognized her and I didn't."
Another name confronted Cain, as well, when one of his two original accusers was identified publicly by news organizations including The Associated Press as Karen Kraushaar, now a spokeswoman in the Treasury Department's office of inspector general for tax administration.
When asked about Kraushaar, Cain said he recalled her accusation of sexual harassment but insisted "it was found to be baseless."