Why not? They are charged with judical power over all cases arising under the Constitution.
Besides, if Oh-One were found to not be a Natural Born Citizen, it would be The People who would force him out, along with anyone from the Congress who rose to his defense.
Nixon was not forced out by the impeachment process, he quit on his own, due to public pressure. His "crimes" were almost insignifigent compared to usurping the very Office of President itself.
Nixon quit on his own because he was going to be impeached. A congressional delegation led by his fellow Republican Barry Goldwater went to the White House to deliver the news that he was finished. It was not public pressure, but political reality.
Had Gerald Ford not pardoned him, Nixon would likely have stood trial for obstruction of justice for arranging to pay off witnesses. Ford sacrificed his own political future to save the country from the continuing spectacle, and I appreciate him to this day for doing that.