"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."
Cain spoke at a hotel on the outskirts of Phoenix, where reporters crowded a small room. Outside, protesters waved signs that read: "Hey Herman. How many more women will you be calling liars?"
There were growing signs of unease in conservative circles as the Georgia businessman tried to stem the controversy in its second week.
"If there is a pattern then it's a part of his character and then, yes, it is going to matter," Tony Perkins, head of the conservative Family Research Center, said in an interview.
An upstart in the presidential race, Cain shot to the top of opinion polls and emerged in recent weeks as Romney's main conservative opponent, with tea party activists and other conservatives flocking to the former pizza company executive's tell-it-like-it-is style and outsider image.
But he's spent the past 10 days battling accusations from women that he acted inappropriately toward them while he headed of the National Restaurant Association in the 1990s.
Cain's campaign issued a tough statement by Tuesday about Bialek, the most recent accuser, including references to civil lawsuits in the Cook County Court system in Illinois allegedly relating to her and cited news reports of her involvement in a paternity case and a bankruptcy filing.
"In stark contrast to Mr. Cain's four decades spent climbing the corporate ladder rising to the level of CEO at multiple successful business enterprises, Ms. Bialek has taken a far different path," the campaign said.
It also questioned whether Bialek had a financial interest in stepping forward.
"Who is financing her legal team, have any media agreed to pay for her story, and has she been offered employment for taking these actions?"
In a round of media interviews early Wednesday, Bialek was asked repeatedly about her motives in speaking out after staying quiet for 14 years.
"I'm just doing this because it's the right thing to do," she said. She said she was neither paid nor offered a job to go public with her allegations. She said she waited so long to come forward because "I was embarrassed ... and I just kind of wanted it to go away."
She said she wasn't paying a fee to Gloria Allred, the attorney whose name has become synonymous with women's rights issues.
At least two women who worked at the restaurant association the same time as Cain filed sexual harassment complaints with the trade group and received financial settlements.
One of them was Kraushaar.
After her name was revealed by several news sites on Tuesday, The Associated Press chose to publish her name after independently confirming that she was one of his accusers. Kraushaar and her attorney previously had attempted to keep her name out of the public discussion, but they issued an anonymous statement last week that confirmed she had complained of sexual harassment and received a financial payout from the trade group. Kraushaar later confirmed to news organizations that she had filed the complaint. And she spoke publicly on Tuesday to The New York Times, saying she had decided to speak out since her name was public.
"When you are being sexually harassed in the workplace, you are extremely vulnerable," she said. "You do whatever you can to quickly get yourself into a job some place safe, and that is what I thought I had achieved when I left."
- Compromise could fix Kentucky's wedding...
- Video: Key and Peele show what it would look...
- Working moms weigh in on Marissa Mayer's...
- Kentucky clerk still won't issue same-sex...
- Ignorance isn't just bliss, it's key to...
- Bishop, Chaffetz say EPA knew spill potential...
- Here's a look at the top 25 best states for...
- Obama gets Iran deal win as Senate Dems amass...
- Kentucky clerk still won't issue... 120
- Compromise could fix Kentucky's wedding... 37
- Latest Clinton emails show frustrations... 26
- In Alaska, Obama depicts stark future... 21
- Obama gets Iran deal win as Senate Dems... 19
- Legal experts see no criminal trouble... 15
- Bishop, Chaffetz say EPA knew spill... 15
- GOP presidential hopeful Rand Paul... 13