Gov. Bobby Jindal of Louisiana called for tougher gun laws in other states on Sunday, breaking his silence on the issue three days after a gunman with a history of mental illness and violence opened fire in a movie theater in the state’s fourth-largest city.

Gun control has become a prominent subject on the presidential campaign trail after the shooting on Thursday in Lafayette became the third mass shooting in six weeks in the United States. Mr. Jindal, who received an A-plus rating from the National Rifle Association, is one of 16 candidates seeking the Republican nomination for 2016.

Law enforcement authorities are investigating how the gunman in last week’s attack, identified as John R. Houser, 59, was able to walk into an Alabama pawnshop and legally buy the Hi-Point .40-caliber handgun he used to kill two women and injure nine other people before killing himself at the Grand 16 Theater. A motive for the shooting has not been determined.

Until Sunday, Mr. Jindal and most of his Republican rivals had deflected questions in recent days over whether the killings reflected a need for tighter gun control laws. On CBS’s “Face the Nation,” Mr. Jindal called for states to adopt laws similar to Louisiana’s that feed information about mental illness into a federal background check system for potential gun buyers.