WASHINGTON — The United States has spent two years chipping away at the World Trade Organization, criticizing it as unfair, starving it of personnel and disregarding its authority, as President Trump seeks to upend the global trade system.

This week, the Trump administration is expected to go one step further and effectively cripple the organization’s system for enforcing its rules — even as Mr. Trump’s widening trade war has thrown global commerce into disarray and another tariff increase on Chinese goods set for next weekend could send markets reeling.

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.