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 off-season to be Rinne's backup, played the final period, making 12 saves.
Anders Nilsson stopped 23 shots for the Islanders.