I think I understand what's going on here.
Your dad thinks that you're required, by law, to keep going and run the dog, or any other animal that gets in your way, over. He simply has a lesser understanding of the law. If you caused an accident by stopping, then you'd have to pay a fine, possibly go to court, etc. But, if you stop when there isn't anyone behind you, and you don't cause an accident, then there is nothing wrong. There is nothing to be at fault for.
Your dad is just a bit confused.