The Constitution is clear on the fact that no Supreme Court Justice can take his seat without the consent of the Senate (barring the interesting case of a recess appointment), but is otherwise silent on what action the Senate might or might not take.
Hence, as far as we know, the Senate is not constitutionally required to do anything at all if the President nominates a candidate they don’t like. They don’t have to discuss it, vote on it, or even acknowledge it, ever. It’s true the rules of the Senate say the President has to re-nominate a candidate if they don’t consider it within some period of time, but of course since the Senate can change its rules any time it likes, this isn’t binding on the Senate itself. It is an interesting question whether they could, constitutionally, confirm a nomination after the nominating President leaves office, dies, is impeached, whatever.
The President’s sole constitutional recourse if he doesn’t like the Senate’s action (or inaction) is to nominate someone else they like better. Conversely, the Senate cannot itself consider a candidate unless and until the President nominates him, no matter how many broad hints they may throw out.
Interestingly, there is a requirement in the Constitution that the President perform his duties (the “faithful execution” clause in Article II, Section 3, as well perhaps as the prescribed form of the oath of his office), but there is no such requirement on the Senate. Hence it may be unconstitutional for the President to refuse to nominate candidates, but not unconstitutional for the Senate to refuse to consider them. An interesting question arises in the case that the Senate refuses to consider a nomination – is the President constitutionally required to nominate someone else after a decent interval? Or can he refuse on the grounds that his first nomination has not been explicitly rejected? (I would guess that if a candidate is explicitly voted down the President is constitutionally required to present another nominee.)