The Constitution only gives Congress the right to remove the president over high crimes and misdemeanors. Ineligibility for office wouldn’t fall under these categories unless Obama committed a crime in the process of declaring his eligibility. The judiciary still has the power to address questions arising under the Constitution. Addressing Obama’s eligibility does inherently mean removing him from office. Judge Carter made an excuse ... a poor excuse for not addressing the questions before his court.
The Constitution allows for the vice president to assume the role of president when said has president has failed to qualify for the position. Said president can also voluntarily step down when it’s discovered he is not actually constitutionally eligible to hold the position he occupied. A court can address this issue without breaching the so-called political question doctrine.