Situations like this,
If someone gave verbal abuse to someone, you'd warn them politely, 1st, 2nd and if they continue to shout verbal abuse, then a 3rd warning comes to action. Whether it was a drunk person, little kid or not.
Though your way of handling the situation....
Anyhow...what's done is done. Yes spitting is a vile thing. Most people don't know how to react in situations like this because it's not common.