"Ain't gonna happen," Cain said a day after a fourth woman, Sharon Bialek, came forward saying Cain made sexual advances toward her in 1997. "I have never acted inappropriately with anyone, period."
Bialek, who held a news conference with her attorney Gloria Allred, alleged Cain groped her in a car outside the National Restaurant Association. Cain, who was president of the organization at the time, was trying to help Bialek find a job.
Cain called Bialek a troubled woman.
"I saw Ms. Allred and her client yesterday in that news conference for the first time. I don't even know who that woman is," Cain said.
Cain told Jimmy Kimmel Monday night that there's no truth to the allegations.
"You get disgusted, you try to control yourself to make sure you watch this thing all the way through, and I was listening very closely, and then when it was all over with I said 'Well, I know what we've got to do,' because there's not an ounce of truth in all of these accusations," Cain said.
Bialek says all she wants Cain to do is to come clean so that people can focus on the real issues. She and Allred were on "Good Morning America" on Tuesday.
"I'm not a liar, and everyone that knows me knows I speak the truth, and that's why I came here," Bialek said. "I wouldn't do something I didn't feel so strongly about."
Bialek was the first to come forward. Karen Kraushaar, who now works for the Treasury Department, has now been identified as accusing him of sexual harassment. She was given a payment by Cain's National Restaurant Association and promised not to talk about the circumstances.
Kraushaar's complaint was found to be baseless, Cain said, but he does not deny the payment.
Cain says he would be willing to take a lie detector test, but he says he won't submit to one without good reason.
Several polls conducted prior to the sexual harassment allegations put Cain in a statistical tie with former Massachusetts Gov. Mitt Romney in the race for the Republican presidential nomination.
Romney called the latest accusations against Cain disturbing and very serious.
A new exclusive Eyewitness News SurveyUSA Poll overwhelmingly shows Californians think Cain is unfit for the presidency: 64 percent say Cain is unfit to be president, 48 percent of Californians believe Cain should drop out of the race. However, 83 percent say Cain's behavior is no worse than other politicians.