SCOTTSDALE, Ariz. -- A defiant Herman Cain declared Tuesday that he would not drop his bid for the Republican presidential nomination in the face of allegations of inappropriate sexual behavior.

"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 said, trying to steady a campaign that has been rocked by the controversy for the past 10 days.

Cain denied anew that he had ever behaved inappropriately, and said of 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 he had never seen Sharon Bialek until she called her news conference Monday in New York.

"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 when one of his two original accusers gave an interview to The New York Times and was identified publicly by news organizations 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."

Cain contended that "the Democratic machine" was pushing the allegations, but said he could not point to anyone in particular. He also suggested that his accusers were lying.

Earlier, Romney, who has been a GOP front-runner for months, told ABC News/ Yahoo! that the allegations were serious "and they're going to have to be addressed seriously."

He called the latest accusations disturbing, and Cain didn't disagree. "He's right. They are disturbing to me," Cain said. "They are serious. And I have taken them seriously."

But they are untrue, he declared.

"I reject all of those charges," he said, adding that "I have never acted inappropriately with anyone."

Cain said it was "a remote possibility" when asked if it were possible that he would recall Bialek's alleged incident in the future.