Over the past two years, Washington has blocked the W.T.O. from appointing new members to a crucial panel that hears appeals in trade disputes. Only three members are left on the seven-member body, the minimum needed to hear a case, and two members’ terms expire on Tuesday. With the administration blocking any new replacements, there will be no official resolution for many international trade disputes.

The loss of the world’s primary trade referee could turn the typically deliberate process of resolving international disputes into a free-for-all, paving the way for an outbreak of tit-for-tat tariff wars.

It could also signal the end of the 24-year-old World Trade Organization itself, since the system for settling disputes has long been its most effective part.

“The W.T.O. is facing its deepest crisis since its creation,” Phil Hogan, the European trade commissioner, told members of the European Parliament this year. If the rules governing international trade can no longer be enforced, “we’d have the law of the jungle.”