Republican fundraiser Elliott Broidy sued Qatar on Monday, accusing the Middle East emirate of trying to discredit him by stealing and leaking emails that detailed his contacts with the Trump administration and Persian Gulf rival, the United Arab Emirates.

The lawsuit, filed in federal court in Los Angeles, adds another layer of intrigue to a complicated narrative that has connected a Middle East dispute to Republican fundraisers, the Trump White House and special counsel Robert Mueller’s examination of Russian meddling in...