It depends on the fridge to be absolutely precise. Some (old) models did not have any electrical control circuitry for operation on gas, and so did not need battery power; you just opened a gas valve and lit the flame and maybe adjusted its size to control how much cooling you got.
Most if not all current models have electronic controls that need 12V power regardless of whether the fridge is operating in gas mode, 120V mode, or 12V mode if it's a three-way fridge. Almost certainly your fridge is in this category and does require some 12V power to operate.