International institutions (especially in the last 30 years) tend to be dominated by countries generally considered to be antagonistic to the interests of the Unites States.
As such, US is rightfully worried that joining an institution which has a binding power over USA without any oversight or veto is bound to end up detrimental to its interests, with US citizens and leaders being convicted on a political ground by a body dominated by forces less interested in promoting justice and prosperity and more interested in "screw the Great Satan".
Given the UN's track record, I don't see such worries as baseless:
United Nations Disarmament Commission reelected Iran to the senior office of a vice chairman. The commission also elected Syria as its recording secretary.
Elected Iran to Commission on the Status of Women
North Korea chaired the Conference on Disarmament
Iran elected as General Assembly vice president.
Saudi Arabia to "U.N. Women's board" agency.
Syria was appointed to two human-rights-related committees of the United Nations Educational, Scientific and Cultural Organization (UNESCO) even as the Syrian government was killing protesters in its own country.
Given that - in the words of an Aussie newspaper covering one of these stories - "lunatics are running the asylim", there's a significant concern that ICC would end up exactly the same.
Considering the repercussions to USA of subjecting itself to jurisdiction of such, it seems significantly more prudent to not join without some guarantee that ICC wouldn't turn into Yet Another OIC puppet.
A separate but related concern is which laws would be applicable.
As just one example, UNCHR seems to greatly care about an all-important right of having a religion not be offended (1 guess which religion's adherents push for that), with "be offended" being in the eye of beerholder. Should that view prevail, ICC might be used to circumvent freedom of speech enshrined in US Constitution's First Amendment.