Answering this question, but purely from its legal and Constitutional aspects and ignoring the political aspects.
And to depersonalize, I will use historical examples.
Most high profile, of course, is Justice William O. Douglas. Douglas took office on April 17, 1939, appointed by Franklin Delano Roosevelt. On December 31, 1974, he suffered a massive stroke while on vacation in The Bahamas. The stroke was centered in the right hemisphere of his brain, paralyzing his left side and inflicting severe cognitive damage. Douglas insisted on attempting to resume participating in court business. The other Justices, recognizing he was mentally incompetent, voted to postpone all business where his vote was deciding. They were finally able to badger him into retiring on November 12, 1975.
But if he had refused, it would have been very difficult to force him to leave and he very well might have caused an obstruction in court proceedings until his death on January 19, 1980.
Because of Justice Douglasâs situation, the Judicial Disability Act passed a few years later permits the President, on the recommendation of the Judicial Conference of the United States, to certify any lower court Judge as involuntarily disabled and allows the President to appoint a replacement for that Judge. But that act does not apply to the Supreme Court.
Imagine Terri Schiavo has been a Justice of the Supreme Court when she entered her persistent vegetative state. Now imagine her bone headed relatives insisted on keeping her alive for 15 years (or more). It would be quite possible that a totally disabled Justice could persist on the court for many years. Probably forcing Congress to add a temporary seat.
Unlikely to be sure, but clearly within the realm of possibility.
Karl Mundt was a United States Senator from North Dakota. Through crippled by a stroke and unable to perform his duties, he refused to resign, effectively leaving his State short of representation in the Senate for three years, until his term expired.
The Constitution is unfortunately silent on the matter of disability.
