Since Obama is not Constitutionally eligible for office, he wouldn’t have to be impeached. Impeachment is for high crimes and misdemeanor. Ineligible is simply that ... ineligible, so yes, the court with the power to resolve controversies that arise under the Constitution could remove an ineligible occupant.
He won the electoral college, he was *CERTIFIED* by congress and sworn in by the CJOTSCOTUS.
*CERTIFIED* would include, the vote tallies where correct, The House and The Senate concurred and approved and deemed him eligible. People forget that Congress has the final say in seating the President Elect and they can and have refused.