GREENWOOD, S.C. — Former Vice President Joseph R. Biden Jr. and an immigrants’ rights activist on Thursday engaged in an unusually extended clash over Obama-era deportation policies in an exchange that turned sharply confrontational. For more than six minutes in a palpably tense gym here during a town hall-style event at Lander University, Mr. Biden sparred with the activist, Carlos Rojas, an organizer with the immigrants’ rights group Movimiento Cosecha. The organization is urging presidential candidates to back a moratorium on deportations on their first day in office and has protested Mr. Biden before. “You should vote for Trump,” Mr....