Like Catholic priests everywhere, Bishop Peter Hickman dons a white tunic each Sunday to celebrate Mass in a sanctuary laden with incense and crosses. Unlike most, he'll often have lunch with his wife and children afterward. "Marriage promotes growth," says Hickman, 50, who has fathered five children, been married three times and divorced twice. "People who've never been married have a hard time knowing themselves." Marriage and children aren't the only things separating Hickman from nearly all Roman Catholic clergy. The church he has pastored for more than 20 years, St. Matthew in Orange, operates much like any other Catholic...