The crisis between the two NATO allies is the worst since the United States imposed an arms embargo on Turkey over its 1974 invasion of Cyprus. But even if Mr. Brunson is allowed to leave, Turkish-American relations won’t improve significantly.

Turkey is holding several United States citizens, including Serkan Golge, a Turkish-American NASA scientist, who was visiting Turkey after the coup and was arrested after a disgruntled relative claimed he had links to coup plotters. Three Turkish nationals working for American consulates in Turkey are also under detention on a cocktail of specious terror charges.

Congress has lined up its own set of sanctions, which include freezing the sale of F-35 fighter jets to Turkey, as retaliation for the imprisonments and Turkey’s plans to acquire the Russian-made S-400 missile system, which the Pentagon sees as compromising NATO security.

The Turkish government’s supporters in the Trump administration argue for de-escalation because they see the relationship with Turkey as too valuable to forsake. They argue that punitive measures would drive Mr. Erdogan fully into the embrace of Russia, China and Iran. And Mr. Erdogan, who is hailed by a large number of Muslims as a fearless champion of the Palestinian cause, could cut Turkey’s ties to Israel. They are wrong.

Should Washington stick to its guns, it might just help nudge Mr. Erdogan back to the democratic path of reform. After all, Turkey has far more to lose from a rupture than the United States does. The Americans don’t have the kind of economic leverage the Europeans have over Turkey. But infuriated by Turkish recalcitrance, Congress now wants the Trump administration to block future funding for Turkey from global lenders like the International Monetary Fund and the World Bank — assistance it might need given its economic troubles.

As Mr. Erdogan continues to systematically hollow out Turkey’s democratic institutions, foreign investors continue to flee. As economic troubles mount, Mr. Erdogan’s sizable pool of supporters in the Turkish business community might rethink their loyalty. Unlike Iran, Turkey does not have oil to subsidize the follies of its leaders.

Despite its cooperation with Russia against the Syrian Kurds, Turkey is nervous about Russian expansionism in the Black Sea and South Caucasus. Turkey does not recognize the annexation of Crimea, champions the cause of ethnic Crimean Tatars and has closed its ports to traffic from Crimea in protest. Mr. Erdogan might talk tough, but Turkey needs NATO and he is aware of it.