Based on the Wikipedia article[0], it seems clear that the Supreme Court can declare a treaty unconstitutional just as it can with a federal statute. The president also appears to have the power to unilaterally withdraw from treaties whenever he wishes and treaties don't take effect without an act of Congress implementing them. In other words, the treaty power is very weak under US law. In short, the US government cannot be bound by any treaty against the will of the people's elected representatives. The fears motivating support for the Bricker Amendment[1] during the Eisenhower administration seem to have been unfounded.
Not talking about that aspect (supremacy of it vs constitution), but this and related stuff with treaties vs federal law:
> The enforceability of treaties was further limited in the 2008 Supreme Court decision in Medellín v. Texas, which held that even if a treaty may constitute an international commitment, it is not binding domestic law unless it has been implemented by an act of Congress or is itself explicitly "self-executing".[26] Law scholars called the ruling "an invisible constitutional change" that departed from both longtime historical practice and the plain language of the Supremacy Clause.[27]
[0]: https://en.wikipedia.org/wiki/Treaty_Clause
[1]: https://en.wikipedia.org/wiki/Bricker_Amendment