"Unanimous" is not the minority.
If the constitution mentioned votes, then the senate would be required to vote. But it doesn't.
Where the Constitution sets out a relationship between two branches, there is a requirement to do something, otherwise one branch or the other could be neutered. If the Senate refused to vote on the president's cabinet, for example, things would be bolluxed. But by your argument, the Senate has the power to do so. Or, if it doesn't do that outright, that it has the power to set the hurdle wherever it wants to.
The constitutionality argument is better fleshed out ...
here -> http://www.freerepublic.com/focus/f-news/1402821/posts
which links to -> http://www.nationalreview.com/mccarthy/mccarthy200505130811.asp