Depp, once among Hollywood's biggest stars, said Heard's allegations cost him "everything." A new "Pirates" movie was put on hold, and Depp was replaced in the "Fantastic Beasts" film franchise, a "Harry Potter" spinoff.
Heard's attorneys have argued that she told the truth and that her opinion was protected as free speech under the U.S. Constitution's First Amendment.
Depp lost a libel case less than two years ago against the Sun, a British tabloid, that labeled him a "wife beater." A London High Court judge ruled that he had repeatedly assaulted Heard.
Depp's lawyers filed the U.S. case in Fairfax County, Virginia, because the Washington Post is printed there. The newspaper is not a defendant.