I had a chance to look at the Hamdan opinions this morning and it looks like my prior perplexity was misplaced. The Geneva Convention holding wasn't independent of the statutory interpretation holding.
Based on my reading of the case (which was both quick and through a dense fog of ignorance regarding international law), here's how the majority got from Point A to Point B. The military tribunals at issue were authorized by Article 21 of the Uniform Code of Military Justice (UCMJ), a federal statute. The Executive sought to escape the requirements of the UCMJ via the Authorization for Use of Military Foce (AUMF) passed by Congress in September 2001 after the terrorist attacks.
Article 21 states that military tribunals must conform with the "laws of war." The Court held that "laws of war" means that the tribunals must conform to the requirements of the Geneva Convention, and that the AUMF didn't abrogate this requirement. So, the right of a private individual to enforce the Geneva Convention in court and the applicability of the Geneva Convention to the Executive's treatment of prisoners at Guantanamo all stem from this single provision regarding military tribunals. In theory, Congress could just amend this provision, or pass a law authorizing military tribunals that don't conform to the Geneva Convention, and the tribunals could proceed apace.
Justice Stevens' majority opinion suggests that the D.C. Circuit's holding that the Geneva Convention isn't enforceable in court is incorrect, but the Court doesn't technically reach the issue since the Geneva Convention analysis arises from the requirements of a Congressional statute. So, it isn't clear that the Geneva Convention applies to every aspect of the conflict with al Qaeda, and this holding could be limited to its facts in the future. Given the fact that this was a 5-3 opinion overruling a decision that Chief Justice Roberts joined when he was on the D.C. Circuit (which is why Roberts didn't participate at the Supreme Court level), I'd say the Geneva Convention won't have much mileage going forward if Bush gets to monkey around with the Court any more by appointing another Justice to replace anyone in the Hamdan majority (Stevens, Kennedy, Souter, Ginsburg, and Breyer).
But the Geneva Convention aside, the key holding was that the Executive exceeded the authority granted by Congress in setting up the tribunals, and therefore the tribunals are illegal. The take-away message from Hamdan is the fact that the AUMF wasn't a "President Can Do Whatever He Wants" resolution by Congress, and the Executive isn't entitled to absolute deference for everything related to the War on Terror. It's now clear that the War on Terror is a joint venture between the President and Congress. Congress is now put to the task of clearly delineating the powers of the President regarding military force, and the President has to stay within the boundaries prescribed by Congress.
Congress' reinforced role in the War on Terror may be a key issue in the mid-term elections (once they take care of more pressing matters like flag burning and the gays). Or, the President could just forget about military tribunals, return Hamdan to indefinite detention, and continue doing what he do without regard for the broader language in Hamdan. The Supreme Court has already blessed the indefinite detention of U.S. Citizens in Guantanamo without charges or trial in Hamdi v. Rumsfeld, nad there's no reason to think the result would be different for foreign nationals. This could be a turning point in America's constant state of war, or just a footnote.
