WASHINGTON (CNN) -- White House Deputy Chief of Staff Karl Rove sharply dismissed an allegation Thursday that he was behind a 2000 rumor that Sen. John McCain was the father of an illegitimate African-American child.

"That is absolutely not true, and I take offense," Rove said in response to a question during an appearance at Troy University in Troy, Alabama. "If you have any bit of evidence that anybody connected with the Bush campaign was involved in that, you bring it forward, because it is a reckless charge." (Watch Video: Rove confronts critic)

The questioner alleged that Rove, then the chief political strategist for presidential candidate George W. Bush, "helped spread the false story" about McCain, who was Bush's main rival for the 2000 GOP nomination.

In the days leading up to the 2000 South Carolina Republican primary, anonymous McCain opponents had spread rumors that McCain had fathered an illegitimate African-American child. At the same time, Bob Jones University professor Richard Hand wrote a widely circulated email suggesting McCain had children out of wedlock. In fact, McCain and his wife had an adopted daughter from Bangladesh. Hand was a Bush supporter but did not have a role with the Bush campaign.

"Do you think people of South Carolina find it attractive to hear that kind of charge made against John McCain," Rove asked at the Troy University event. "Or do the people of South Carolina respond to it as they should have, 'What a remarkable thing that John and Cindy McCain adopted a child from Asia, took him into their home as an act of compassion and kindness.'"

"The Bush campaign had nothing to do with it, and the Bush campaign endeavored to stamp out those kinds of things because they hurt George Bush and helped John McCain, not the other way around," Rove added. "Either I'm a genius, or I'm an idiot. Only an idiot would spread trash like that and expect to do their candidate any good."