The Constitution says that the Senate can set it's own rules, and one of those rules states that it takes 60 votes to shut off a debate. And that applies to all debates, not just judicial nominations. Now if the Vice-President wants to rule that the Democrats are in violation of Senate rules then that's one thing. He'll have to cite the Senate rule that they're in violation of. But if the Vice-President wants to shut down the Democrat filibuster on the grounds that it is unconstitutional then that's way beyond his authority to decide.
The AP article on Fox is misleading.
The plan is for a senator to raise a point of order to close debate on a nominee.
The presiding officer, VP Cheney, would sustain the point of order. The minority would appeal the ruling.
The senate would then vote to table the appeal, which can be done with 51 votes. End of filibuster on this nominee.
The Constitution doesn't enter into it at all because neither the point of order nor the Chair's ruling has anything to do with it.
BTW, this is not a precedent setting scenario. During the '70's and '80's, Sen Robert Byrd used the same tactic four times to alter Senate rules. With 51-vote majorities.
This plan also does not require a change in Senate Rule XXII, which governs the filibuster. If the GOP has the party discipline to give the motion to table 51 votes, then they will not have to do it again and nominees will get their up-and-down votes.
The Constitution says that the Senate can set it's own rules, and one of those rules states that it takes 60 votes to shut off a debate. And that applies to all debates, not just judicial nominations.
As I understand it each incoming senate sets its own rules. All that is needed is a new rule that says a simple majority is all thats needed to shut off debate. I may be mistaken, but if not I say they should do it.