The related question, Could the President abolish the Supreme Court?, addresses the extreme situation, whether a President could realistically shirk his Constitutional duty to appoint new SCOTUS justices indefinitely until all sitting justices had died/retired, and by so doing indefinitely prevent the Court from hearing or deciding any cases.
This question is more narrow and more practical. With Scalia's death, Obama is seemingly bound by the Constitutional definition of his office under Article 2, Section 2 to appoint a replacement. However, Senate Majority Leader Mitch McConnell has said that he believes the next President, not Obama in his lame-duck last year, should choose Scalia's successor, and has made it clear he intends to block the confirmation process of any Obama SCOTUS appointee any way he can.
The question is simple; does President Obama even have the option of not nominating a replacement, leaving that task to his successor as McConnell says he should, or does Article 2, Section 2 require him to at least put forth a nomination before he leaves office? Obama himself has said he does not believe he can do anything other than appoint someone.