The Constitution typically is interpreted to be superior to every branch of government in the United States. It lays the framework for the Federal government and grants its powers while laying out explicitly several restrictions.
Article VI of the Constitution includes what is called the "Supremacy Clause":
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, any Thing in the Constitution or Laws of any State to the Contrary notwithstanding.
Where the Constitution lays out that it, along with the laws passed by Congress and signed by the President and any treaties ratified by the Senate, shall be the "supreme Law of the Land", the Constitution is establishing its supremacy over all acts of the States, but only insofar as the Constitution grants Congress the power to do such actions.
But the Supremacy Clause tends to be misinterpreted by those advocating the separation of church and state. I’ve heard an argument similar to this on multiple occasions: the Supremacy Clause in effect provides that the language of a treaty shall be legally in equivalence to the language of the Constitution. Most recently I heard such language in a new Constitution lecture by Shane Killian on the separation of church and state.
In this lecture, at around 7:44, Shane makes this argument with regard to Article 11 of the Treaty of Tripoli:
According to Article VI [of the Constitution], any treaties made in pursuance of the Constitution are part of the Supreme Law of the Land. This declaration therefore has the same effect as if it were part of the Constitution itself.
In the comments to his video, I argued that he was "a little off" with this argument, that if the Constitution grants any treaties any legal equality to anything, it’s statutes enacted by Congress. He responded by posting Article VI to the Constitution, following with "Direct quote", as if that were to mean anything.
In his response, I quoted directly from Reid v. Covert, 354 US 1 (1957): "This Court has regularly and uniformly recognized the supremacy of the Constitution over a treaty".1
To firmly show that the Constitution is superior to any actions of Congress, and to further display that treaties are equivalent to other ordinary actions of Congress, the Court further declared in Reid:
This Court has also repeatedly taken the position that an Act of Congress, which must comply with the Constitution, is on a full parity with a treaty, and that when a statute which is subsequent in time is inconsistent with a treaty, the statute to the extent of conflict renders the treaty null. It would be completely anomalous to say that a treaty need not comply with the Constitution when such an agreement can be overridden by a statute that must conform to that instrument.2
Further, it is very dangerous to state that treaties have "the same effect as if it were part of the Constitution itself". This implies that treaties can bypass the amendment process laid out in Article V, and that treaties can confer additional powers to the government, as if ratifying a treaty is, in effect, amending the Constitution without going through the necessary process. A very dangerous notion indeed, and one that the Supreme Court also directly addressed in the Reid decision: "no agreement with a foreign nation can confer power on the Congress, or on any other branch of Government, which is free from the restraints of the Constitution."3
The Constitution is supreme to any action of the Federal government, and this includes treaties. This was first stated by the Supreme Court in Marbury v. Madison, 5 US 137 (1803): "Thus, the particular phraseology of the Constitution of the United States confirms and strengthens the principle, supposed to be essential to all written Constitutions, that a law repugnant to the Constitution is void".4 In Marbury the Supreme Court overturned an act of Congress that sought to modify the Supreme Court’s jurisdiction, which is firmly established in Article III, Section 2 of the Constitution.
Again the Constitution is supreme to any act of the Federal government, including treaties. The Treaty of Tripoli or any treaty to which the United States is a party is not equal to the Constitution in any way, and to say that a treaty is equal to the Constitution is dangerous.