In the above quote, it sort of sounds like O'Connor is trying to contrast treaties and the U.S. Constitution, as if a treaty can usurpt the authority of the Constitution.
The power of entering into treaties is part of the Constitution, and any aspect of a treaty must be Constitutional and it must be ratified into law. The contrasting "but", in the above O'Connor quote makes it sound like O'Connor believes that treaties can override the U.S. Constitution.
Concerning Breyer, his remarks now explain his past rulings and the combination of his remarks and the past rulings now makes him deserving of impeachment.
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This Constitution, and the Laws of the United States which shall be made in Pursuance thereof; and all Treaties made, or which shall be made, under the Authority of the United States, shall be the supreme Law of the Land and the Judges in every State shall be bound thereby, any Thing in the Constitution or Laws of any State to the Contrary notwithstanding.
I think this is the clause they intend to use to hold treaties equal in law with the Constitution. As I read it treaties are superior to State law based on this clause. That alone is scary stuff.