Trump appeared to base his argument on the clause in the Constitution that calls for removal of a president "on Impeachment for, and Conviction of, Treason, Bribery, or other high Crimes and Misdemeanors," claiming "there are no Crimes by me at all."
A unanimous Supreme Court in 1993 concluded that impeachment by the Congress is a "political question" that is nonjusticiable -- which means not reviewable by the courts. The case involved two federal judges who had been impeached by the House and convicted in the Senate, but the ruling is seen as having wider application across government, including the presidency.
The court held that the Constitution confers on the Senate the "sole" power to try impeachments.
"A controversy is nonjusticiable where there is a textually demonstrable constitutional commitment of the issue to a coordinate political department," Chief Justice William Rhenquist wrote for the court.
He noted that Article I of the Constitution "demonstrates a textual commitment of impeachment to the Senate."
It's also unclear what grounds the president would use to bring an impeachment case before the Supreme Court in the first place, especially because Trump wrote that he would do so immediately after hearing the Democrats were moving forward on impeachment.
"There would be an additional problem beyond the  problem, which is that such a lawsuit would be stunningly premature," said Steve Vladeck, a constitutional law professor at the University of Texas School of Law. If impeachment proceedings had yet to occur, it would be unclear "what the president was even challenging," Vladeck added.
But aside from the nuance of Trump's timing in going to the Supreme Court, Vladeck emphasized that Rhenquist's majority opinon "goes out of its way" to say the court does not want to get involved in impeachment proceedings, in no small part because of what might happen specifically if the proceedings target the president.
"If the House votes to impeach a president and the Senate voted to remove him, that should be the end of the matter," Vladeck said. A lawsuit at the conclusion of presidential impeachment proceedings could raise a host of new questions, he said, such as who would hold the office of the presidency during the litigation.
"It's supposed to be the end of matter, and not beginning of the lawsuit," Vladeck said.