Rinne played two periods in his first game since left hip surgery in early May.
Injuries limited Bourque to 34 games for the Predators last season.
Watson, a first-round selection of the Predators in 2010, saw his first NHL action last season and scored one goal for Nashville.
Carter Hutton, signed in the offseason to be Rinne's backup, played the final period, making 12 saves.
Anders Nilsson stopped 23 shots for the Islanders.