First, find out why he doesn't want to be there. I've found out (from working with a lot of young kids) that they'll blow hot and cold when it comes to sports. One day they're crying because they don't want to go, the next week they LOVE it and don't want to leave. It's something with the ages 5-8. They're ALL like that. They get upset easily over small things and then they don't want to go and they'll act like they will hate it FOREVER.
Then you find out it's because one of the kids or coaches said something. I had one 6-year old say she wanted to quit gymnastics because I let her do a trick 2 times, but the girl before her got to go 3 times. It's usually stuff like that!
If he won't say why, take him to the next game and try again. Yes, the spitting is uncalled for but c'mon, anyone who works with kids has to expect the unexpected. I wouldn't think poorly of a kid forever because they spit on me. Six year old are unpredictable. If it was a reoccuring thing then maybe I'd worry, but a one time thing is not really a big deal.
Basically I'm saying that six-year olds are fickle, so try again. Just because he didn't want to be there one day doesn't mean you should pull him for the season.
(I will also mention that one of my six-year olds didn't want to perform at a tumbling competition. After having her cry about it, we gave her back to her mother and suggested they take her home this time. We find out later that she was upset because she didn't get a chance to finish watching her cartoon because they had to leave for the competition. This made her so mad, that she decided she HATED competing and didn't want to do it that day. Her parents solved the problem by not letting her watch TV before competitions. She is now 9, and still loves gymnastics.)
You punished him for the spitting, so no need to quit over it. He should apologize to the coach, and then continue on.