Let's be careful, here. Our beloved Constitution says that any treaty made under US authority is the supreme law of the land.
Article VI, Sec 2: "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, anything in the Constitution or laws of any State to the contrary notwithstanding."
That last part is a the most troubling. You know that future Leftists will use this to say that International Law binds US law, no matter what the Constitution says.
Woe to he who would knowingly disarm his brother to lead him to death.
"If an act of the legislature, repugnant to the constitution, is void, does it, notwithstanding its invalidity, bind the courts, and oblige them to give it effect? Or, in other words, though it be not law, does it constitute a rule as operative as if it was a law? This would be to overthrow in fact what was established in theory; and would seem, at first view, an absurdity too gross to be insisted on." -- Chief Justice Marshall.
re: "anything in the Constitution or laws of any State to the contrary notwithstanding."
That last part is a the most troubling. You know that future Leftists will use this to say that International Law binds US law, no matter what the Constitution says.
That is their plan, although it relies upon a deliberate lie and intentional misreading of what the US Constitution actually says. It's been a while since I studied this, but I believe we have Allen Dulles, brother of John Foster Dulles to thank for this nasty and dangerous bit of sophistry.
The entire phrase "anything in the Constitution or laws of any State to the contrary notwithstanding." refers only to the laws and Constitutions of the various states. In other words, (x + y)*Z = xz + yz. (Constitution or laws)*any State = State Constitution and/or State laws.
In cases where the Framers were referring to the federal Constitution of the United States, they specified so explicitly or used the phrase "this Constitution" to differentiate it from other Constitutions. Hence the preceding phrase Article VI, Sec 2: "This Constitution...". It they had been referring to the federal Constitution and had intended that treaties could trump the US Constitution itself, then that last phrase would have read thusly: "...anything in this Constitution or laws of any State to the contrary notwithstanding."
Again, I agree with you, the Left is going to make this argument - we just need to remember that it is both logically invalid according to the actual text of the US Constitution, and it is a lie.