The argument wouldn't be that the House doesn't have the right to impeach. It would be on the interpretation of "high crimes and misdemeanors."

Article II, Section 4:

The President, Vice President and all civil Officers of the United States, shall be removed from Office on Impeachment for, and Conviction of, Treason, Bribery, or other high Crimes and Misdemeanors.

Short of being impeached explicitly for treason or bribery, the administration certain could challenge that the offense meets that standard, and that would involve the Court. I'm not at all saying I think that challenge is valid or would hold up, but it would invalidate the statement that "the Supreme Court has nothing to do with impeachment."