That's exactly the rule that Cheney would declare unconstitutional under the Senate rule that allows him to. That is exactly what the "constitutional" and/or "nuclear" option is. He would declare that rule can't apply to appointments that the constitution declares requires 50% plus 1 to pass.
But VP Cheney (and you) would be wrong. The Constitution says no such thing.
The Constitution says that appointments require the consent of the Senate, and the Constitution also says that the Senate makes its own rules.
The Senate could require unanimous consent for nominees without violating the Constitution.
The Vice President can facilitate, from the chair, changing the cloture rule by 51 Senators. I think he should do so.
He cannot invent Constitutional mandates where none exist, which is what you would have him do.