The claim that the USA's opposition to the ICC is a necessary and justifiable result of upholding the US constitution is nothing but a quite transparent lie. A couple of reasons:

  1. The court will not admit cases which are being investigated or prosecuted genuinely by a state which has jurisdiction over it (Article 17, 1(a) of the Rome Statute). Thus, the USA could easily prevent any American from ever being tried in the ICC by trying them under their own oh-so-great system of justice instead. But I guess what they really want is to reserve the right to not even try war criminals, as long as they're American war criminals.
  2. If concerns about procedure or yet-to-be-defined crimes were the issue, the US could easily demand clarification and define clear requirements that would have to be met in order for the US to support the ICC. If those were resonable and fair, I'm sure the international community would be willing to accomodate. However, so far the USA have insisted on its citizen to be immune from prosecution, period.
  3. It is perfectly normal and accepted that if you leave your country and commit crimes elsewhere, you will be tried under the laws of the country where you committed them. The USA isn't willing to go to war about its citizens being tried for murder or drug trafficking elsewhere. Apparently because those crimes can usually not be tracked back to the US government itself.
  4. The US government cares not a nickel for the constitution and the benefits of its judical system when they apply to people it doesn't like. POWs of the war against the Taliban are deliberately kept off US territory in order to avoid having to grant them all those pesky rights they would have if they were tried in a US court. Better yet, the US claims they're not even POWs at all, because POWs have some rights too.

So to summarize: There is only one reason why the US government is opposed to the ICC, and it has nothing to do with standing up for any principles or rights, except the principle that The USA can do no wrong (because "wrong" is defined as "what the USA doesn't like") and the right of the USA to do whatever it damn well pleases, in any way it pleases. And if that includes genocide, crimes against humanity or a war of aggression or two, then so be it.