Technically he is not allowed in the main casino. However, I have seen many kids much younger than 16 being allowed to watch their parents play. From my observations, as long as he is quiet and doesn't draw attention to himself, he will probably be okay. If he looks a few years older than his age, he may even get away with playing.
Also, depending on the ship, there may be places where he can "legally" watch you play. For example, I was on one ship (the Regal Princess, I believe) that had a small lounge with a small dance floor and bar right within the casino. There was usually music and dancing here in the evenings. The kids were allowed in this lounge area, which was separated from the casino only by a metal railing. Some of the slots and video poker machines were within a foot or so of the lounge seating, making this the perfect place for him to watch from.
Another example is the Sapphire Princess. On this ship, there are several video poker machines located in the back of Club Fusion. Club Fusion is a fun, lively lounge where alot of the late night entertainment (such as karaoke and the 50's dance party) takes place. Kids are allowed in when accompanied by an adult. My nieces would often sit and watch my parents play video poker while waiting for the entertainment to start.
Other ships may have similar areas where he is allowed to be while still having a good view of the games.