After he repeated the sentiment in a tweet, Mr. Cain responded, “My AR is ready for you Robert Francis,” referring to Mr. O’Rourke’s given name.

The exchange underscored the tensions surrounding gun control, a highly politicized issue that became an especially personal one for Mr. O’Rourke after a shooting on Aug. 3 killed 22 people in El Paso, his hometown.

Since then, Mr. O’Rourke has reset his campaign and made gun control a central platform in his bid for the presidency.

His aggressive plan to combat gun violence includes a national firearm buyback program, which would be mandatory for assault-style weapons under penalty of fine, and voluntary for handguns.

He would also require universal background checks for gun purchases; create a national gun registry; limit the influence of the National Rifle Association by banning political action committee contributions to members of Congress; and ban silencers, bump stocks and high-capacity magazines.