Can the President direct the states to follow a ruling of the World Court on the meaning of of a U.S. treaty?

This is the subject of Medellin v. Texas, argued — with great vigor — in the Supreme Court yesterday. SCOTUSblog summarizes:

With cross-currents of constitutional and international law flowing freely, what appeared to be a majority of the Justices looked askance at a Presidential memo in February 2005, directing nine U.S. states to give 51 Mexican nationals convicted of crimes in those states a new chance to test their rights under an international treaty, the Vienna Convention on Consular Relations. What was troubling those Justices the most, it seemed, was that the President had sought to make binding a ruling by the World Court that would otherwise not have controlling effect on states’ ciminal [sic] procedures. That was worrisome for two reasons: it might intrude on the Court’s role to say what the legal meaning and effect of treaties is, and it might empower the World Court, in effect, to dictate the substance of American law....

Justice Anthony M. Kennedy ... twice raised concerns about whether the President could “displace the authority” of the Court to interpret judgments of the World Court. ... The World Court ruling itself, [Medellin's lawyer] said, is federal law — binding on the states by virtue of the Supremacy Clause, and also binding because the President had concluded through his Article II power to it was in the nation’s interest to comply with the World Court judgment.

... [Solicitor General Paul] Clement stressed that the government did not support Medellin’s argument that, without the President’s action, the World Court decision would be binding on the state courts in the U.S. “The President’s role is critically important,” the Solicitor General argued....

Two members of the Court who often talk of the value of looking to foreign law for guidance, Justices Stephen G. Breyer and Ruth Bader Ginsburg, were clearly the most willing on Wednesday to show support for the World Court’s authority and the duty of the U.S. to fulfill its promise of obeying World Court judgments in interpreting the Vienna Convention. Thus, they were the most energetic questioners of Texas’ state Solicitor General, R. Ted Cruz.

Justice Breyer sought Cruz’s response to a simple “chain of logic,” going from the language of the Constitution making treaties “the law of the land,” the U.S. agreement to the Vienna Convention and its agreement to abide by World Court rulings applying that treaty, and then to the requirement of the World Court that states take full account of the Convention’s strictures. Cruz said that Texas did not dispute that treaties were the law of the land, but said that what was at stake here was the action of the President in seeking to make the World Court judgment binding on the states in contradiction to their own laws. The Convention, he said, is not a self-executing treaty, and the President cannot make it so on his own....

This is an extremely difficult question. Here's the PDF transcript of the argument. I will write something more about it later.