A push by the casino billionaire Sheldon Adelson to outlaw online gambling has ignited a bitter civil war in the gambling industry, dividing one of Washington’s most powerful interest groups and posing a major test of the Republican megadonor’s political clout. Mr. Adelson’s effort officially kicked off on Wednesday, when lawmakers, including a senator, Lindsey Graham of South Carolina, who has accepted tens of thousands of dollars from the businessman and his family, introduced legislation originally drafted with Mr. Adelson’s lobbyist. The bill would close a three-year-old loophole in federal law, banning online gambling — a growing industry that Mr....