The UN always was a system by which to use treaty law to get around the Constitution. The fix was in since before the Constitution was signed.
He could try what GWB did when he ordered the state of Texas to grant a new trial to a Mexican convicted of raping and murdering two teenage girls and sentenced to death, because he didn't see a Mexican consular official (Medellin v. Texas). When the case got to the SCOTUS, even John Paul Stevens voted against Bush.
But now that Kagan replaced Stevens, maybe Obama will try to foist "international law" on us again.