The deepening standoff raised questions of whether the two strong-willed leaders were risking even broader chaos as they vied for the upper hand in a widening diplomatic dispute largely focused on individual personalities.

An evangelical preacher, Mr. Brunson, who has lived in Turkey for 23 years, is one of about 20 Americans, including a NASA scientist and chemistry professor from Pennsylvania, who have been swept up in Mr. Erdogan’s crackdown since the failed coup two years ago.

Mr. Erdogan says the coup was orchestrated from the United States, and specifically by a Muslim cleric, Fethullah Gulen, from his self-imposed exile in Pennsylvania. Turkish authorities have demanded that he be extradited, something American officials have dismissed, and the detained Americans are widely seen as bargaining chips.

Mr. Erdogan has shown no sign of backing down. He sounded defiant in two speeches to supporters on Friday, railing against foreign powers that he accused of orchestrating Turkey’s economic crisis, and vowing not to bow to Western pressure.

“Those who believe they can make us kneel by economic manipulation, have never understood this nation at all,” he told a crowd in Gumushane, a province on Turkey’s northern Black Sea coast. “They can’t use threatening language, blackmailing language against this nation. Especially bullying will not cut it with our nation.”

After watching Mr. Erdogan on live television, Asli Aydintasbas, senior fellow with the European Council on Foreign Relations, said, “They have no deal, their positions have hardened.”

The breakdown was surprising, she said, because Mr. Trump had appeared to reach the outlines of an agreement for the pastor’s release with Mr. Erdogan on the sidelines of the NATO summit last month.