According to a new paper by Illinois Law professors Robin Kar and Jason Mazzone, President Obama has the constitutional power to appoint—and not just nominate—a replacement for Justice Scalia. They write: 

After Justice Antonin Scalia’s recent death, politicians wasted no time before teeing up a political battle over his replacement. Republican Senators—led by Senate Majority Leader Mitch McConnell—immediately announced that they will not consider or vote on any replacement nominees from the current President. In doing so, they have taken a position that may be constitutionally problematic in ways that have not yet been fully appreciated. Now that President Obama has nominated Judge Merrick Garland to the Supreme Court, this problem requires greater attention.

The crux of the problem is that an outright refusal on the part of the Senate to consider any nominee from President Obama arguably works a delegation of an elected president’s Supreme Court appointment power to an unknown successor. While the Appointments Clause of the Constitution allows Congress to delegate a president's appointment power in certain instances, it does not permit delegation with respect to Supreme Court appointments. Hence this delegation raises a potential problem of separation of powers. Historical practice also cautions against any effort to delegate to a future president the authority to nominate and appointment a member of the Supreme Court.

There are 104 prior cases in which an elected president has faced a vacancy on the Supreme Court and began the appointment process prior to the election of a successor. In all 104 cases, the sitting president was able to both nominate and, with the advice and consent of the Senate, appoint a replacement justice.

Full paper on SSRN