A minority DOES pass legislation.
For example, last night I was watching the end of the session with my wife, and I said "Now watch this, dear".
THen Frist, with only the President and a minority representative present on the floor, went through a series of bills.
For each bill he asked the the bill be presented, then asked that they be considered as passed, with unanimous consent.
So in fact multiple pieces of legislation were passed last night with NOBODY voting, and with one senator asking for it.
But that is because the rest of the senators AGREED to let that happen.
So yes, the requisite number of senators (for example, 2/3 for impeachment) could vote to assign a committee to decide, and vote to accept thier decision as final. There would be nothing unconstitutional about it, because it would have been a 2/3 vote.
If the constitution mentioned votes, then the senate would be required to vote. But it doesn't.
Look at the Congressional records from the beginning. The actual votes and conclusion (passed or not passed) are all there.
"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