OK, I'm going to start by asking our members not to be too hard on you, but I'm sure they're going to want to flame you, I feel the same way.
You're lucky to have your dog with you at all. A dog that bites a person, and doesn't have a current rabies vaccination, can be euthanized right away, because the only way to determine if an animal has rabies is looking at it's brain tissue.
I suspect your vet is correct, the antibiotics probably made him sick, and the rabies shot can also upset their stomach. My 13 year old just got revaccinated for rabies Tues, and she had loose stool yesterday.
So, I think the likelihood of your dog having rabies is very slim. I believe the symptoms take a couple weeks to develop anyway. However, you put your dog at terrible risk, by not having him currently vaccinated, putting him at risk for both rabies and legal repercussions, and also, you risked him when you brought him to the dog park. I hope he recovers quickly, and I hope you make more responsible choices for your dog, and for the breed you represent.
How is the other dog? Was anyone hurt seperating them.