I had the very same thing happen to my retired greyhound but on a larger scale. He lost all of his toenails and only had the tip of the bones exposed.
The vet won't do all that much to help you out. Treatment for this is cleaning & dissinfecting the paw regularly, 2-3 times a day for the first 3-5 days and then once a day for a week or so after that.
It will heal up and a nail will start growing again but not like it used to be. My greyhound is doing absolutely fine now and his paws don't bother him at all. Try not to worry too much, it's not too serious of an injury.
To help prevent this from occuring again, be sure to trim your dogs nails regularly, even if it's just a little bit. Once a week is the norm.
Good luck and keep us posted.
P.S. You can dissinfect with peroxyde, baxedin and other gentle antiseptics.
Keeper of 5 snakes, leopard geckos, 1 green iguana, 20+ tarantulas, 2 dogs & a bunch of rats!