My baby is six months old. Every time we put him on his stomach, after a few minutes he throws up. He gets very frustrated on his stomach. He doesn't roll over by himself yet.
Many babies don't like being on their stomachs, mainly because they can't see very much and babies rely heavily on their vision for early development.
In fact, it's often because they are frustrated in this position that they push hard to get their heads, then shoulders, up and off the surface to get a better view. Once they get high enough, and twist around to follow a sight or sound, they lose their balance and roll over from front to back. This helps to develop strength and coordination of their arms and upper bodies.
So, while your son might not like being on his stomach, this 'tummy time' is still a good idea, even for 20 minutes or so a few times a day. As for the spitting, this is most likely to happen within 30 minutes of a feeding, so timing this activity around feeds should help.