For background on the ICC, see Rome Statute of the International Criminal Court (Wikipedia).

Article II, Section 2 of the Constitution states:

[The President] shall have power, by and with the advice and consent of the Senate, to make treaties, provided two thirds of the Senators present concur;

In practice, President signs UN treaties and submits them to Congress for ratification.  In some cases, the treaty can be left in limbo if the former happens but not the latter, such as the forty-year lag on the ratification of the Convention on the Prevention and Punishment of the Crime of Genocide.   The Rome Statute is an even stranger case because President Clinton signed it in 2000, but chose not to even submit it to the Senate for ratification, stating:

The United States should have the chance to observe and assess the functioning of the court, over time, before choosing to become subject to its jurisdiction. Given these concerns, I will not, and do not recommend that my successor, submit the treaty to the Senate for advice and consent until our fundamental concerns are satisfied.

Concerns about his successor proved unwarranted.  Section 9 of the 2002 National Security Strategy (NSS) went a step further:

We will take the actions necessary to ensure that…[“we”] are not impaired by the potential for investigations, inquiry, or prosecution by the International Criminal Court (ICC), whose jurisdiction does not extend to Americans and which we do not accept.

The curious thing about this is that numerous public opinion polls and surveys have shown that Americans support the ICC by wide margins.  In particular, a 2008 poll (pdf) by the Chicago Council on Global Affairs found that

There is also bipartisan support for the International Criminal Court (ICC), with  68 percent of Americans saying the United States should participate in the agreement on the ICC that can try individuals for war crimes, genocide, or crimes against humanity if their own country won’t try them.

So while its jurisdiction may not extend to Americans, it’s certainly not accurate to say that Americans wouldn’t accept it if given the choice.  Nevertheless, the farcical American Service-Members Protection Act (Hague Invasion Act) was passed in 2002 as well.  The Act finds that

Any American prosecuted by the International Criminal Court will, under the Rome Statute, be denied procedural protections to which all Americans are entitled under the Bill of Rights to the United States Constitution, such as the right to trial by jury.

Which brings us to the drone strikes.  The drone strikes/assassinations are very illegal in Pakistan and the Obama administration has formally declared (the Bush administration did so implicitly) that it will use them on whoever they want, including American citizens without “impairing” itself with the whole procedural protection deal that was the rationale for obstructing the ICC.