At a news conference in Phoenix, Cain said he doesn't even remember Sharon Bialek, the latest woman -- and the only one to identify herself publicly -- who accused Cain of groping her in a car more than a decade ago.
The alleged harassment occurred while Cain headed the National Restaurant Association in the 1990s.
"I tried to remember if I recognized her and I didn't. I tried to remember if I remembered that name and I didn't," Cain said. "These charges and the accusations I absolutely reject. They simply didn't happen. They simply did not happen. …
"I have never acted inappropriately with anyone. Period. …
"These anonymous accusations are false," he said, adding that his wife Gloria remarked, "That doesn't even sound like anything you would do to anyone."
Asked by reporters if he'd be willing to take a polygraph test, Cain said yes if there was a good reason to do so.
Cain described Bialek as a "troubled woman" and accused Democrats of trying to undermine his campaign. Last week he blamed rival Texas Gov. Rick Perry's campaign.
"Sexual harassment is a very serious charge. In no way have I tried to minimize sexual harassment in the workplace," Cain said, adding that he has acted in the past to rectify such situations.
Cain characterized the accusations as a smear campaign designed to get him to drop out of the race. That won't happen, he said.
"For decades the American people have wanted a businessman in the White House, not another politician. … When a businessman by the name of Herman Cain stepped forward [those in power set out to destroy his candidacy]. We are not going to allow Washington or politics deny me the opportunity to represent this great nation. As for these accusations leading me to back off and withdraw from this presidential primary race, it ain't going to happen."
Cain dismissed the allegations of a woman "who actually filed charges" as a personnel matter in which no actual violations were found. The woman has been identified as Karen Kraushaar, a Treasury Department employee. She told The Washington Post she never wanted to go public, but she is willing to now.
Bialek went on national television earlier Tuesday to say she spoke out to help Cain's bid for the White House.
"I actually did it because I wanted to help him. I wanted to give him a platform to come clean, to tell the truth," Bialek said Tuesday on CBS' "The Early Show." "And he still hasn't done it and it's really a shame because he could have."
Bialek, 50 and a Republican, said she wouldn't rule out voting for Cain if he admitted what he had done.
"I'd have to think about that," she said. "I hope that he does and I'd have to think about that one."
In a news conference Monday, Bialek described an incident that allegedly occurred in Washington when she asked Cain to help her in a job search after she lost her post with the trade group's education foundation.
Bialek told reporters Cain reached under her skirt for her genitals and "grabbed my head and pushed it toward his crotch" while in a car after a dinner meeting to discuss her job search.
After she expressed surprise and shock, Bialek said Cain responded, "You want a job,right?"
Cain has denied the accusations ever since they first surfaced last week. Cain Monday told late-night talk show host Jimmy Kimmel Bialek's story was a "totally fabricated" and "there is not an ounce of truth to these allegations."