Our unleashed dog ran across a street by the park, and was hit by an oncoming car. The driver did not stop and kept on driving. There was a truck behind the car who saw what happened, and the truck sped up and caught up with the driver who hit our dog, and brought him back. The driver who hit our dog said that he did not see the dog, and now not only does he not want to be responsible even partially for the vet bill, but he is demanding payment for his damaged car.
First, we are sorry to hear about your dog’s injury, and hope that they make a full recovery!
This sort of situation depends largely on three things: 1) the actual facts, 2) witnesses to the facts, and 3) various laws in your jurisdiction. Do you have the contact information for the driver of the truck, and are they willing to be a witness? What are the leash laws in your jurisdiction? For example, if you were in violation of the leash laws, you may be fully liable for any damage done to the car, and they may have no liability at all for your dog.
You may want to start by asking your vet whether they are familiar with these sorts of situations, and what they recommend. Also ask your vet if they can recommend an attorney who is familiar with these situations.
The bottom line is that you will need to consult with an attorney familiar with dog-versus-car situations.
In the end, if the leash laws of your jurisdiction do not absolve the driver from having hit your dog, you may want to look into filing a small claims action to recover the cost of veterinary services. However, you may also want to negotiate with the driver if it seems you may be liable for damage to their car. If they are willing drop their damages claim if you drop your veterinary claim, you may find it less costly to pay the vet, and not have to pay for car repairs. Again, it all depends on the laws where you are.