It is an extension of that idea of veto authority that the administration is insisting on for U.S. personnel, most of them military, doing peacekeeping duty. The U.N. Charter was designed to give to the major players on the world scene in 1945 absolute power to shield themselves from U.N. over-reaching. Parliamentary efforts were made (indeed, led by the United States) to diminish that authority by the device of denominating certain actions as administrative rather than substantial, thereby devolving authority to the General Assembly where there is no veto. But the veto survives as an expression of the limits a major country is prepared to go to in submitting to others' political authority.
Since 1945, the United States has grown hugely in relative power; indeed, it is the only superpower. This means that we uniquely can have our way by force of arms, but it means also that commensurate obligations are imposed on us. In economic trade, we agree to acknowledge the authority of the World Court to opine on differences, but we are at liberty to defy the court's judgments. In war and peacekeeping, the position of the government is that no international court may assert jurisdiction over an American citizen. One reason for it is that sassing Big Brother is a quite universal temptation. There are here and there (most notably in Spain) jurisprudential exhibitionists who would love to haul in the American general in charge of bombing weddings in Afghanistan and Chinese embassies in Belgrade.
It's not a bad idea to keep on talking about how to devise better means of guarding against jurisprudential opportunism. We acknowledge the authority of most laws of most other countries (we would agree to jailing an American who stole in Saudi Arabia, but not to chopping off his hand). In 1951 we passed the Status of Forces Agreement with the NATO countries, and have relied, in principle, on the proposition that American soldiers must be held accountable when violating the laws of countries in which they are doing military duty.
We are not alone in holding out on the ICC. Russia has not ratified the treaty, and it hasn't even been signed by China or India. The word to get out to our allies is that their primary concern should be to effect U.S. cooperation in peacekeeping exercises. And that means reassuring Congress, which finances and ultimately authorizes such ventures, that Americans aren't going to be held up by a foreign court.