The Constitution of the United States does not empower the Supreme Court to decide who can be the president.
From an intellectual standpoint, it is not hard to get yourself so tangled up in being questions that you become paralyzed. However, if you insist on turning this whole thing into a philosophical question:
Right now, Barack Obama either is or is not the president of the United States.
If Barack Obama is the president, the Constitution grants to the House of Representatives the "sole" power of impeachment and to the Senate the power to remove following impeachment. The Constitution does not provide the Supreme Court with that power.
If Barack Obama is not the president, then there isn't any need for anyone to do anything. You can just ignore what everyone tells you about the so-called "President Obama." You're not required to believe any of it.
And yet, this is not borne out in reality: what happens if you are a soldier who refuses to abide by orders due to the superiors having no authority? {The chain of command is dependent upon the fundamental qualities of authority: the one who is sent is under the authority of the one sending; this goes to the Commander-In-Chief, which the Constitution assigns as the President.}