The president has a political duty to nominate someone to fill a vacancy on the Supreme Court and the Senate has a political duty to consider it.
But the Constitution draws no distinction between the President's role in nominating Supreme Court justices and the PResident's role in nominating the Assistant Secretary of Labor for Veterans' Employment and Training (or any other of over 1000 "officers of the United States" that are appointed by the President with the advice and consent of the Senate.
The distinction is practical and political -- the practical, and thus political, consequences of not filling a vacancy on the Supreme Court are greater than not filling most other vacancies. But from a Constitutional perspective,they are the same.
"[The President] shall have Power, by and with the Advice and Consent of the Senate, to make Treaties, provided two thirds of the Senators present concur; and he 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: but the Congress may by Law vest the Appointment of such inferior Officers, as they think proper, in the President alone, in the Courts of Law, or in the Heads of Departments."