The head of the FBI on Tuesday recommended against charging Hillary Clinton for mishandling classified information while she was secretary of state, even as he harshly criticized the presumptive Democratic presidential nominee for being “extremely careless” with the nation’s secrets.

In a 15-minute statement at Federal Bureau of Investigation headquarters, James Comey said “any reasonable person” in Mrs. Clinton’s position should have known the sensitive material involved merited greater security. The announcement ended a...