Under what clause of the Constitution?
SCOTUS is a court like any other. It does not make pronouncements unless a case is brought before it. It then rules on the case.
SCOTUS' only power over Congress is the Constitutional duty to rule on the Constitutionality of laws passed by that body and signed into law by the Executive. But, it can only do that if a case is brought before it challenging the constitutionality of the law. SCOTUS chooses those cases carefully, most rulings allowing the judgements of lower courts to stand.
The College of Electors now has the power, at its discretion, to document eligibility. It has not chosen to do so, because no law requires it.
The US Military cannot constitutionally act without Congress, or Congress and the Executive's authorization. The "US Military" is not empowered to decide anything. The President is Commander in Chief. If a case were brought before the courts, and the President were found finally to be ineligible to have been a candidate, it would not then be the court that removed him from office, but the Congress.
Our best, and probably our only, bet for removing Obama is that he is ruled ineligible for 2012 by a state or states that makes documentary proof of eligibility mandatory before placing him on that state's ballot, or giving him that state's electoral votes.
If he accepted that state's ruling, there would be no reason for any other state to accept him as eligible. If he did not accept the ruling, he could challenge it in court, finally giving the courts the ability to hear the case, finally deciding what a "Natural Born Citizen," is, etc..
If Obama has stacked the court by then, it is entirely possible that they will interpret Article 2 in a way I guarantee we will not like.