Mitch McConnell, Majority Leader of the Senate, says the President should not appoint a successor. But actually, it is a Constitutional duty of the President:
"He [the President] . . . shall nominate, and by and with the Advice and Consent of the Senate, shall appoint Ambassadors, other public Ministers and Consuls, Judges of the supreme Court, and all other Officers of the United States, whose Appointments are not herein otherwise provided for, and which shall be established by Law . . ."
-U.S. Constitution, Article II, Section 2
So in urging the President not to make an appointment, McConnell is actually urging the President not to fulfill a duty affirmatively commanded by the Constitution.