Cain: "I have never acted inappropriately with anyone, period."

"The charges and the accusations I absolutely reject: They simply didn't happen. They simply did not happen."

"As [Sharon Bialek] got to the microphone, my first thought was, 'I don't even know who this woman is,' " said Cain. He said he could not remember Bialek from the National Restaurant Association, where he was president and CEO from 1996 to 1999, and "where she said she worked."

Asked about Karen Kraushaar, an accuser whose name was made public Tuesday, Cain called her allegations baseless. "To the best of my recollection that is the one that I recall that filed a complaint, but it was found to be baseless." He said he doesn't remember any of her accusations, except for making a gesture that she was the same height as his wife.

Cain said there was a "machine" trying to keep a businessman out of the White House, and said Sharon Bialek was a "troubled woman" put forward by "the Democrat machine."

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