The general answer is that independent courts which have the power to prosecute and publicize injustice and are potentially equipped to enforce justice; that such courts protect the weaker ones from the stronger ones, on all levels of the justice system.
The stronger ones on all levels have less incentive to support justice since
- they can protect themselves;
- the privilege of behaving as they see fit, without the fear of being prosecuted, appears advantageous.
If you want a more abstract formulation, you could say that transferring some sovereignty — clearly a disadvantage — to supranational bodies makes only sense (in a purely economical sense, and perhaps on short time scales) if the benefit surpasses the loss. This is the case for small countries whose effective freedom to move internationally is limited to begin with, but who are under constant threats from stronger players.
By contrast, the loss-benefit saldo is probably negative for strong nations who can move at will on the international stage and have next to nothing to fear from other nations (or only from other nations who themselves have limited respect of international institutions).
This rationale applies without reservation to the United States' apprehension towards international institutions, of which the International Criminal Court is just one example.
Your question may relate to the — perhaps Hegelian — idea that the system of law is an expression of a collective consensus on a common subset of ethical principles. The law's institutions therefore would be seen as indirect guardians of ethics, through the enforcement of law. This is actually nicely expressed in the English term "justice" which does not distinguish between ethics and law: Merriam-Webster gives, among others, the definitions of "the quality of being just, impartial, or fair" — an ethical category — as well as "the administration of law", a category in the realm of of government.
Consequently, an entity refusing to subject itself under the law and its agents could be seen as rejecting to subject itself under collectively aggreed upon principles of ethics, in other words: One could legitimately assume that this entity, man or nation, reserves the right to act unethically.
Would such a stance not contradict the United States' spirit and principle of the promotion of human rights? I suppose that this perceived dissonance is underlying your question.
It is easily dissolved by realizing that between the United States' words and the United States' deeds lies and always has lain a chasm which becomes wider and deeper the closer one looks. This is obvious from the perspective of these deeds' victims; victims in other parts of the world and victims within the country, insofar as they are still alive. For the Americans who benefit, or believe to benefit, the chasm is bridged by a mixture of conscious deception and propaganda on one hand — cui bono? — and naive ignorance on the other.