From what I understand, one objection is that trying an American citizen in an international court violates the right to due process. How? There presumably is a process defined by a treaty which the US has agreed to.

The other objection: ...implies that the Court does not have the power to try Americans for crimes committed on U.S. territory.

Is it not in Congress' power to grant jurisdiction to a court?

Edit: (http://en.wikipedia.org/wiki/Supremacy_Clause) This Constitution, and the Laws of the United States which shall be made in Pursuance thereof; and all Treaties made, or which shall be made, under the Authority of the United States, shall be the supreme Law of the Land; and the Judges in every State shall be bound thereby, any Thing in the Constitution or Laws of any State to the Contrary notwithstanding.