Ah, the heartbreak of parenthood (I'm serious.) You have asked what you can do. You say your son offered a half-hearted apology. If you want him to learn why his behavior hurts others as well as himself, he'll have to do a lot better than half-hearted apologies.
Most people believe a sincere apology is enough. I believe firmly in restitution accompanying an apology whenever possible. Since nothing concrete was broken here, restitution is trickier.
You have offered an apology, but you aren't the guilty party. Your son did the damage to the fellow he spit on. His parents hurt for their son. What you feel is probably exponentially more than anyone else.
First, your son must apologize properly for what he's done. This is best done in front of the parents, for they were hurt as well. To apologize effectively, he should
1. Own the offensive behavior (it didn't just happen; he doesn't get to say, "I don't know why I did it..."), which means he has to think it out honestly. You can help him do this.1
2. Verbalize why it was wrong (he needs to think with empathy about how this might have affected the other boy. This means he doesn't get to say, "It wouldn't have bothered me," or, "I would have laughed.") How would your son have felt - honestly - if the reverse had happened? Belittled? Humiliated? Disrespected? Loved? Befriended? Happy? To discuss this, he needs an emotional vocabulary2. If he is not naturally empathetic, you'll need to work on empathy with your son.
3. Express sincere regret for having caused the boy... (humiliation/embarrassment/etc. - all that apply.)
4. Promise the behavior will not be repeated.
5. Ask if there's anything he can do to mitigate the hurt he's caused. (E.g. if he did this in front of other kids, he should also apologize publicly as well. This is mitigating if the recipient was humiliated.)
6. Act on his apology.
How does he make restitution for his offense to the other party? Should I try to make amends, more than the apology I've already offered?
The restitution here is something that would restore the recipient's sense of self-worth. That's pretty difficult, but a sincere apology does help towards that end; it means he cares about the boy's feelings.
He started at a new school this year and has been acting out excessively. I'm scared he's becoming a loner at school because he plays rough.
This is a very valid concern. Kids who felt like outsiders during their childhood often carry the effects of this for a very long time.
As a parent, you need to find out why he's acting out. You don't mention anything that might point to a cause of this except for being at a new school. If that's all that is different, it does make it easier.
If there's more (he's at a new school because there was a separation and you moved; he's at a new school because there was a job change and you moved; he's angry at losing his old friends, there are cliques at his new school, he's in a very different kind of school environment in which he feels insecure, etc., etc.) then your job is harder, but whatever the case, you need to start having some regular talks about why before you can effect a change.
If it's nothing that he can identify after trying his hardest, he may just be less strong in reading social cues than his peers, making him act inappropriately towards them. In that case, sincere apologies wherein he needs to practice empathy will help him understand the effects of his behaviors, as will working on empathy at home. He might also benefit from exercises in reading social cues (i.e. how to make friends, or at least stop losing them.)
You might try reading articles by Michelle Garcia Winner whose work is to help children learn how to act acceptably towards others (much of her work is free online.) After you've gotten a good idea of the principles, you can talk to your son's teacher(s) and the school counselor (if they have one) for their assessment of his behavior. Then you can start working with your son at home.
Hopefully, this will be all that's needed. If your son didn't have problems before this year, loving him, working on empathy, and working on social skills with him will likely help a lot.
You have my best wishes.
1 "I thought it was funny," while true, is also not self-reflective enough. What did he hope for? The boy to laugh? To elicit laughter of the other boys? Why did he want this? Was it a way to curry favor? Popularity? As I said, this will take a lot of discussion, best done in session lengths that exceed his comfort level (this is serious) but not to the point of non-productivity.
2 A feeling wheel can help here. It's not too advanced. There are a lot of resources online for teaching (and modelling) an emotional vocabulary.