Actually, it's as opposed to being born not entitled to US citizenship and then later becoming a US citizen via the process of naturalization. For instance, Schwarzenegger, Granholm, and Kissinger are ineligible because they are naturalized. They were born in Austria, Canada, and Germany, respectively, and only later became US citizens.
The court could agree that it meant born of US parents but no judge is wanting to take that step for fear of igniting riots.
No judge has the power to take that step. Under the Constitution, the sole power to remove the President resides with two thirds of the Senate, acting on a motion to impeach passed by a majority of the House. The most the Court could do is cough up an opinion (and send John Roberts, in extra-fancy robes, down to the Senate to count the votes).
The Court would not be ordering the removal of a President, but rather a usurper. They would first be saying that he was never elibigle, and then if necessary and probably in a latter case, order his removal. Not that it's likely to be necessary, if the Supreme Court should ever declare him to be ineligible. Someone would either persuade him to step down, or it would be torches and pitchforks time. (although I'd pretty a nice rifle myself). If he's ineligible, who is going to defend his sorry ash, the military? Nope their oathers are to the President, not some resident usurper. The Secret Service? Same for them, they defend the President.
The Constitution provides that the President may be removed by impeachment and conviction in the Senate, it says nothing about removing a usurper. I think the founders figured the Second Amendment covered that potentiality.