People in Europe are learning right now how not awesome it is to get yourself into binding international frameworks with people who are connected by incomplete legislative and executive jurisdiction, and whom often have dramatically different national priorities and values.
In any case, it's arguably unconstitutional for the U.S. to support international courts. If Congress signed up for a treaty binding us to the decisions of say the IJC or ICC, would those decisions be reviewable by the Supreme Court? If so, then accepting jurisdiction would be redundant, since the Supreme Court is perfectly competent to adjudicate disputes under "international law." If not, that would arguably be a separation of powers violation: Congress signing us up to accept the superior jurisdiction of a foreign court whose judgments couldn't be reviewed by what is supposed to be the court of last resort: the Supreme Court.