The problem is the Supreme Court can legally ignore the issue if it is coming up from the lower courts or the National Grand Jury plan. They can ignore lawsuits from servicemembers and regular citizens.
BUT Article III, Section 2 of the U.S. Constitution
“In all cases affecting ambassadors, other public ministers and consuls, and those in which a state shall be party, the Supreme Court shall have original jurisdiction.”
What that means is that if the State of Texas or Alaska for example legally challenges the qualification of Obama to hold office that the case doesn’t start out in some lower court with most likely a Clinton appointee shooting it down like they tend to do. It is heard by the Supreme Court directly.
If Obama refuses the Court request for evidence of his proper birth as a citizen then they could null and void his actions as President and more importantly he would no longer be the legal Commander in Chief. It would legally allow the US Military to remove him from office if force of arms was neccessary.