Although I'm not a politician, I may be able to provide some insight.
Some recent developments notwithstanding, the US is generally very keen on limited government, separation of powers, checks and balances, and all that. They're generally rather paranoid about a government abusing its power (and indeed, they're often right to do this, as even the US government's history of abuses is not small in the least).
The ICC, at least as currently envisioned, does not guarantee those being tried the same rights that the US Constitution does for its citizens. Indeed, from an American perspective, there are some very severe flaws; see my writeup on the Statute of the International Court of Justice for more. There is no presumption of innocence, no jury trial, no appeal, no double-jeopardy protection, and a whole host of other problems.
Ashcroft's statements about terrorists notwithstanding, the US generally believes that all people should have the same rights that it guarantees its citizens. Of course, as a nation, it can only effectively protect the rights of its own people, but at least it can do that. As far as that goes, I can't say I disagree with the spirit behind the law. The US government has a duty to protect the rights of its people, as all governments do. In this case, it has no choice but to oppose the ICC, which can take those rights away.