"We believe this is an isolated incident, a domestic-violence dispute," he said.
Father Robert Milbauer, the parish's pastor, said a woman injured in the attack was the gunman's ex-wife. The two have a child who attends the school and had been mired in an ongoing dispute, Ruiz said.
The gunman's identity was not immediately released.
He opened fire with a .22-caliber rifle a few minutes before the church's annual weekend festival was to begin. Milbauer said about 50 people, mainly church volunteers and their children, were setting up food and game booths and carnival-style rides when the gunfire erupted.
"I was walking toward the festival area to say an opening prayer and I saw them," Milbauer said of the shooting victims.
The man's 30-year-old ex-wife was one of the festival workers. She was hospitalized in stable condition, Ruiz said.
A 45-year-old man was shot in the chest and was in critical condition and another man, 47, was in stable condition with a leg wound, Ruiz said.
Their identities were not immediately released.
The man walked away after the shooting but was quickly grabbed by bystanders, one of whom was an off-duty Burbank police officer.
"They managed to overtake him and held him down," Ruiz said. "I'm told that he was in the process of possibly reloading."
The festival was shut down for the day, and Milbauer said grief counselors were meeting with witnesses, particularly the children.
The parish plans to go ahead with the festival on Sunday, Milbauer said, in part to help parishioners put the tragedy behind them.
The church and school are located in the city's Granada Hills area in the San Fernando Valley, an ordinarily peaceful, multiethnic, middle-class residential neighborhood not far from the historic San Fernando Mission.
The annual festival is held to raise money for the school's building fund.