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.

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