Actually, it does. The Supreme Court has held that all "acts of Congress" must be in accordance with the Constitution. Ratifying a treaty is an act of Congress. Thus, if Congress ratifies a treaty that imposes requirements on the US that violate the US Constitution, the court can hold that ratification to be invalid.
I think the point the parent was trying to make is that what's legal in the US doesn't make a whit of difference to how other countries treat us.
In other words, we can't force other countries to not all agree to raise tariffs on goods that they import from the US. The constitution doesn't matter, because they are foreign governments. Our laws have no bearing there.
If that wasn't the case then any country could use their laws to force people in other countries to do whatever. Certain activities would be both required and banned at the same time. Here in the US it's illegal to coerce a voter, and in Australia voting is mandatory. There's no way to reconcile those two laws.
Thus far, no actual treaty has ever been invalidated; the closest is an executive agreement in Reid v. Covert: https://en.wikipedia.org/wiki/Reid_v._Covert