Interesting, now that you mentioned it, I looked up the vote and saw that it was only 54-45. Maybe there still is hope. In any case, though it still required at least a majority of the Senate to approve. 2/3 of them aren't going to turn around and impeach him for it.
Why NAFTA wasn't taken to court over this bogus distinction I'll never know.
I don't know what difference it would make. The government will just say that it's not a treaty; it's an act of Congress. It's not like the courts have recognized any meaningful limitations on those.
The government would be lying. When a nation "agrees" with any other nation, the laws of nations are very clear that such an instrument is a treaty. Further, the Congress and the Administrative branch often lack the Constitutional authority to execute the terms of extranational rulings.
Yeah, I know, that doesn't stop them.