This is exactly what I was going to post but you already have.

You raise an interesting idea however... A third party would be required for this to work but in keeping with the bitcoin spirit you could have that third party on your phone.

In essence your ring stores some sort of key which can be given to a person in order to request a payment (which includes the amount and the public wallet address to send to). The key the ring shares is simple to authenticate the request. This request could be sent to your smart phone which then notifies you requesting you to enter a pin to authorize the payment. The app on your phone then automatically makes the correct transfer out of whatever account you have set it up with. Your private key is never exposed. The only downside is that previous keys you share could be used to make additional requests. This could be solved by you periodically updating the verification key in the ring.

That should work... And I don't see any obvious security flaws in it (but I may be wrong). Implementation wise however the vendor who you are trying to pay will need to know how to request a payment and the format that it needs to be in (assuming that we can't make a standard) I'm not sure how much data can be stored in the ring. But storing all that might be a bit too much.

It really would be so much easier if the ring could do the Authentication inside it. However this more convoluted method does have the benefit that even if an evil party had access to your ring, they would still need your phone AND pin number. I guess that's about as much as you can hope for.

EDIT: After sleeping on it, If you have to have your phone with you to approve transactions it would be far simpler just to have your phone scan a QR code or interact through NFC to perform the action. The ring adds nothing and makes things a good deal more complicated. You could potentially set up a remote server to perform the same features but at that point I think you have moved beyond what is viable for the general public.