Is my vet out of line? Putting my deaf dog down?

File a harassment suit against the vet. And a restraining order. Then, find a new vet. No vet would put a dog down for merely being deaf. If you're dog is well behaved and that vet is actually worried about his well being, he would ask you to sedate Rapture before bringing him in for appointments. In my opinion, your vet probably wants to take your dog to sell him to a testing facility looking for deaf dogs to try out new devices on. No vet would offer you money otherwise.