The President clearly has the right to nominate a replacement. It’s more ambiguous when the Senate needs to do their part, and they can always reject the nomination as the Democratic controlled Senate did to Robert Bork.
Both sides are going to try to leverage the situation politically where they can, but if the arguments get too bogged down by Constitutional law the average voter will tune out and fall asleep. Thus far the American people haven’t seemed especially angry when either party uses the filibuster to stall anything from happening.
