In November 2003, Raymond Burke, then the bishop of LaCrosse, Wis., instructed priests in his diocese to deny Communion to three politicians unless they publicly recanted their pro-abortion rights positions, an action some Catholic scholars say is tantamount to excommunication. Since then, Burke's supporters have increasingly pointed to what they see as parallels with another case, and another moral hero, from 40 years ago. In the wake of the United States Supreme Court's 1954 Brown v. Board of Education ruling, Joseph Francis Rummel, Archbishop of New Orleans, began planning the integration of the archdiocese's schools. It would take eight years,...