A bill introduced in the Senate on Tuesday would require law enforcement to get a search warrant before obtaining old data like emails stored on servers -- a major update to U.S. digital-privacy laws.

The bill, by Senate Judiciary Committee Chairman Patrick Leahy (D., Vt.), would amend a 1986 law called the Electronic Communications Privacy Act, which extended restrictions on government wiretaps to data transmissions as well as phone calls.

Critics...