CEDAR RAPIDS, Iowa—Feuding escalated Friday between the leading Democratic presidential candidates, with former Secretary of State Hillary Clinton saying Vermont Sen. Bernie Sanders has been cowed by the gun lobby and calling on him to admit a mistake by casting votes that she said were favorable to the gun industry.

Mr. Sanders, while campaigning in Iowa, sought to put Mrs. Clinton on the defensive on the question of how best to help those who need a paid leave of absence from work in the event of family crises. He chided...