Here's a link to the opinion (89 page pdf).
I haven't read the entire opinion yet, but here's the holding from the syllabus:
"Held: Neither Avena nor the President’s Memorandum constitutes directly enforceable federal law that pre-empts state limitations on the filing of successive habeas petitions."
Johnathan Adler at Volokh writes:
"Chief Justice Roberts wrote the majority opinion, which held that neither a judgment of the International court of Justice nor the President's executive order directing state courts to follow the ICJ's judgment constituted federal law that pre-empts a state's pre-existing bar on the litigation of subsequent habeas petitions. Justice Stevens concurred in the judgment, while Justice Breyer wrote a dissent on behalf of himself and Justices Souter and Ginsburg."
There's some great links from that Volokh page, too.