What law said that the President could not fire anyone unless Congress approved?

False

That was one of the main topics behind the impeachment of President Andrew Johnson in 1868. Although U.S. Supreme Court Chief Justice Salmon Chase, who presided over the impeachment trial in the U.S. Senate, did not directly address the question, the U.S. Supreme Court later stated that the President may, indeed, fire someone who a President had appointed with the advice and consent of the U.S. Senate.