Well he was very wrong. A president has the right to nominate someone until the day they leave office, no matter what party the presidency is occupied by.
Agreed. And the Senate has the right to reject that nominee for any reason.
This entire affair is an example of the Republicans being their own worst enemy. They never should have boasted that they wouldn't consider the nominees. They should have heard them out and then either voted them down or not given them a vote. Then the spectacle would have been Obama demanding his nominee be accepted despite voting down both Bush nominees.