The medical examination will take place at the government hospital in Kolkata on Thursday, the Court said, and the report has to be submitted in a week. The West Bengal police chief has been asked to set up a team to help with the medical assessment.
Justice Karnan faces contempt after he named 20 "corrupt judges" of the country earlier this year and sought an investigation alleging corruption in the judiciary.
When he appeared in the Supreme Court in March, Justice Karnan refused to respond to the contempt notice and later went on to "summon" all seven judges in a "hearing" that he held at his Kolkata home.
Since then, the 61-year-old judge has matched every Supreme Court order on him with his own, announced during what he calls are "residential" court hearings.
At a hearing on March 31, Chief Justice JS Khehar had raised questions about the judge's mental health.
Last week, Justice Karnan asked that the chief justice and seven Supreme Court judges not be allowed to travel outside the country. The order, he said, would not be cancelled until the cases against him were scrapped.