Justice Scalia was an American hero. We owe it to him, & the Nation, for the Senate to ensure that the next President names his replacement.— Ted Cruz (@tedcruz) February 13, 2016
There is, quite simply, no legitimate basis to obstruct any and all possible nominations by Obama to fill Scalia's seat. The Constitution on this is clear: the President nominates, subject to the Senate's "advice and consent." It does not place a hold on the President's power when elections are imminent or the President's successor has been elected. And to assert that the Senate may justifiably use its "advice and consent" role in order to block any nominations would be to invite jurisprudential chaos. Such an abuse of power would be subject to no limit - conceivably, the Republican Senate could just as easily have promised to block any Supreme Court nominee from Obama's re-election.
And to rely on the Senate's rules to delay and block nominees - these are not Constitutional mechanisms. If they are abused as Cruz, et al., promise, the Republican Senate will threaten to push Obama to assert that his constitutional prerogative is being unconstitutionally violated by the Senate - the result being, perhaps, a constitutional crisis the likes of which this country has never seen. Because, crucially, the only remaining branch able to resolve the dispute would itself be the subject of it.
Nothing new, to be sure, for the Republicans. They have been at the forefront of violating the Constitution while claiming to respect it.