You could can create a non-counterfeitable coin by embedding an NFC chip that will sign a message using the private key.
Now that is an interesting idea. Pondering it now...
Passive chip with enough crypto to sign a message with the private key.
Hmmmm....
Still have not plugged the hole in all physical coin mechanisms: you have to trust the maker to not make duplicates or keep copies of the private key for themselves.
But assuming a reputable coin maker ala Mike C (may his business RIP for now) you might be on to something.