A defiant Herman Cain declared Tuesday 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 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."

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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."

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

Earlier, Romney, the former Massachusetts governor who has been a GOP front-runner for months, told ABC News/Yahoo! 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, both in an earlier interview and at the news conference.

"He's right. They are disturbing to me," Cain said. "They are serious. And I have taken them seriously."

But they're untrue, he declared.

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"I reject all of those charges," he said, adding that "I have never acted inappropriately with anyone" and didn't even recognize Bialek.

Though recent polling shows Cain still doing well, party operatives suggested it was only a matter of time before his political standing could suffer.

"Herman's base is going to stick with him," said Republican strategist Rick Tyler, Newt Gingrich's former spokesman. "But the average Republican voter who is not as engaged as intensely in the race, is sick of this and, for Cain, the concern is they will pass on it and pass on him."

Cain looked to keep those supporters in his corner.

"We are not going to allow Washington or politics to deny me the opportunity to represent this great nation," he said.

"As far as these accusations causing me to back off and maybe withdraw from this presidential primary race? Ain't gonna' happen. Because I'm doing this for the American people, and the children and the grandchildren."