If the dog ran out suddenly in front of your vehicle and there was nothing you could do to avoid the accident then you are not liable for damage to the dog. However, if the dog was just standing there and you could have avoided hitting it without causing harm to yourself or other traffic, then you are arguably liable for damages to the dog. These are obviously two ends of the spectrum, so honestly in your heart of hearts determine where your facts fall. The closer you are to the second scenario, the more I could see a small claims court judge ordering you to pay.
This answer is provided for informational purposes only. Actual legal advice can only be provided in an office consultation by an attorney licensed in your jurisdiction, with experience in the area of law in which your concern lies.
If the dog darted in front of your vehicle, you would have no legal obligation to pay for the vet bills. As a driver, you must act as a reasonably prudent driver would act in a similar situation. Generally, it is not foreseeable that a dog or any other animal will cross your lane of traffic.
In addition, most municipalities will have leash laws that require all dogs to be restrained when outside of an enclosure. The law is to protect the public and the dog from injuries.
If you are sued, you should contact your insurer immediately.
I agree with my colleagues - most places have leash laws which means the owner would be liable to damages to your car; however, as has been pointed out, if you could have avoided hitting the dog, the analysis may be different.