BERLIN—The German government on Thursday issued an unusually strong rebuke of U.S. intelligence activities here, warning that spies who break German law would be prosecuted and that continuing revelations of American surveillance were damaging ties between Washington and Berlin.

The government’s statement came after Chancellor Angela Merkel’s chief of staff, Peter Altmaier, met with U.S. Ambassador John Emerson at the Berlin chancellery to discuss newly leaked documents that appeared to show American surveillance of senior...