[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[.]
At some point, most presidents have probably gotten annoyed by something the Supreme Court did. Could the President decide to stop appointing Supreme Court judges, in violation of this duty ("shall"), even to the point where no judges remain?
This seems like it would be harder to do legally than the Senate abolishing the Court (see question on that over here).
Alternatively, could the President satisfy his duty while still reaching the same end by nominating only candidates who (s/)he knows will never be confirmed (e.g. because they are nowhere near qualified)?
Could is intended to refer to legally could; political feasibility considerations would be at Politics, not Law.
A much narrower related question: Could President Obama do as Sen. McConnell says and leave the SCOTUS nominations to the next President?