We have the Uniform Code of Military Justice for military personnel, and on paper, a Congress to hold the President accountable, although that obviously needs to be beefed up because the results lately have been unsatisfactory. It does not need to be the ICC, nor should we allow it.
I don't understand what you mean by "allow it". The ICC are free to do what they like, and the US are free to ignore it (outside any sanctions the international community impose).
And then we have presidents that almost uniformly pardon people convicted of the grossest war crimes.
Trump pardoned Clint Lorance who ordered murder of civilians. Before that, William Calley convicted of multiple murders had his sentence commuted by Nixon to 3 years of house arrest.
