Ronald Reagan didn't have a formal public debate with Tip O'Neill. Bill Clinton didn't have one with Newt Gingrich. Even if you'd become the head of the Republican Party the old-fashioned way, i.e., through electoral politics, you wouldn't have any reason whatsoever to expect the president of the United States to directly debate you. When they're not running for office, presidents don't do that -- presidents you like don't do that, presidents you don't like don't do that. So, sorry, but your idea is idiotic.
OOPS: I guess Kevin K. said more or less what I just said hours ago....