Perhaps Reynolds, a law professor at the University of Tennessee and a graduate of Yale Law School can help me out. I searched the document for relevant passages. I found the following:
Article I, Section 5, Clause 2: "Each House may determine the Rules of its Proceedings . . ."Note that it specifies a two thirds vote to approve a treaty, but not to approve judges and other officers appointed by the president. I assume that since the second clause I quote is silent on the matter, the first controls and the Senate can make its own rules which currently provide for filibusters when 60 senators don't vote to cut off debate.
Article II, Section 2, Clause 2: "[the president] shall have Power, by and with the Advice and Consent of the Senate, to make Treaties, provided two thirds of the Senators present concur; and he shall nominate, and by and with the Advice and Consent of the Senate, shall appoint Ambassadors, other public Ministers and Consuls, Judges of the supreme Court, and all other Officers of the United States, whose Appointments are not herein otherwise provided for, and which shall be established by Law . . ."
The good professor should explain why it is unconstitutional for the requirement to apply to judicial nominees, but not other votes.
Update: I just noticed the post below the one that Reynolds linked to. It describes the "plain words of the framers" saying that "51 votes to confirm a nominee is all that is required." Of course, the "plain words" say no such thing, especially since there would have only been 26 senators at the time of the founding.
You have to actually read the constitution sometimes.
UdateII: George Will comes to the same conclusion on the Constitutional issue and quotes from article I, section 5, clause 2 of the Constitution as well. I guess you aren't required to have read the Constitution to graduate from Yale Law School, or to teach it at the University of Tennessee.