WASHINGTON—Iran has released U.S. Navy veteran Michael White from prison on a medical furlough to the Swiss embassy, the State Department said Thursday, a response to a soaring rate of new coronavirus infections the country.

Mr. White, 48 years old, was released on the condition he remains in Iran. The Swiss Embassy has represented U.S. interests in the country since 1980. A statement by his family welcomed the decision.

“We...