Wolfowitz made his statement on Monday to a panel investigating whether he broke bank rules when securing Riza's rise. She is expected to appear later in the day.
Wolfowitz said the issue had become "circus like".
"I, for one, would not give in to such tactics. And, I will not resign in the face of a plainly bogus charge of conflict of interest."
"I do not believe that doing so would serve the interests of the world's poor who are supposed to be the first concern of us all."
Wolfowitz has been criticised by Europe since he took the job in 2005.
"Only when the cloud of these unfair and untrue charges is removed, will it truly be possible to determine objectively whether I can be an effective leader of the World Bank," he said.
Almost simultaneously at a news conference in the White House Rose Garden, George Bush, the US president, reiterated his support for Wolfowitz, but said the issue had not been discussed while meeting Angela Merkel, the German chancellor and EU president.
Bush said: "I think he ought to stay. He ought to be given a fair hearing."
"Wolfowitz must resign because ... his act of favouritism for personal interest was wrong"
Ibby, Mumbai, India
Send us your views
The bank's 24-member board is expected to make a decision in the case this week, which has also raise expressions of concern that the bank's mission of helping the poor would be damaged.
Wolfowitz said he regretted the "tumult" the controversy had caused the bank, but said: "To criticise me when I did nothing other than attempt in good faith to follow the guidance of the ethics committee would be unwarranted and grossly unfair."
"Moreover, it would be harmful to the institution," he said.