You need to check with your local/state association.
I would advise you to have game administration deal with him as you are correct that he is not part of the team currently playing.
I had a similar situation come up out here a few years ago.
During the girls game the boys team was sitting in the first couple rows of the stands on the opposite side from the team benches and the table.
During the first half and following a call one of the boys threw his baseball cap onto the court. I had game administration remove him from the gym.
The boys coach came over before the start of the 2nd half and asked if that meant that his point guard couldn't play in the next game. I told him that the kid could play, but that he'd better be very well behaved.
I checked with my state association later, and they agreed that this was the proper course of action. In order to support my decision, I argued that the NFHS rules are only rewritten for single games. The rules are not designed to govern multiple contests. Those kind of regulations must be put in place by the local league/state.