EL CAJON, Calif.—The backyard gathering was part Catholic Mass, part rebellion.

The priest, an Iraqi immigrant, had been kicked out of the local church. Parishioners had been warned by local church leaders not to worship with him. Yet 50 people sat in makeshift pews behind a home east of San Diego in a show of opposition to church officials urging Christians to stay in Iraq, where their numbers are dwindling.

“There...