The Evangelical from North Carolina has been in detention for nearly two years, accused of having links to banned political groups.
However, President Trump demanded his immediate release, or Turkey, a NATO ally, would face “large sanctions”.
The Turkish Foreign Ministry called the U.S. demands “unacceptable”.
Spokesperson Hami Aksoy said no-one can give orders to Turkey and threaten his country.
Pastor Brunson is a long-term resident in Turkey.
He lived with his wife and three children while working as the pastor of the small Izmir Resurrection Church, which had a congregation of about two dozen protestants.
The authorities accuse him of having links with the outlawed Kurdistan Workers’ Party and the Gulenist Movement, which Turkey blames for a 2016 failed coup.
Mr. Brunson has denied charges of espionage, but faces up to 35 years in jail if found guilty.
He was moved into house arrest on Wednesday for health reasons, but US Secretary of State Mike Pompeo said this was not enough.