I am sorry your dog ate your nuvaring. He may be able to pass it without any problems, but I am concerned about a reaction to the ring and any medicine it may contain. I think it would be best if you called the SPCA's animal poison control hotline to determine if you need to seek veterinary care or not. See the following website for the phone number.
Time is always of the essence in these situations, and the longer it takes to seek veterinary care, the more difficult and expensive it can be to manage the problem. The ring might pass with no problems, but why wait until it gets stuck? Why take that chance that will require emergency surgery? If there has not been too much time since the ring was ingested, the vets might be able to pull it out with an endoscope, which is less expensive and far less invasive than surgery. I really think you should call the hotline or head straight to an emergency vet, right away. Please let me know if I can be of further assistance. If this has been helpful, please hit the green accept button. Best of luck with your handsome man, I hope he does well :)