Not normally, no. It is possible to get your dog back though, if you have the Knothole Island DLC. On Knothole Island, there is a tomb in the graveyard, which has a dog statue on it and a lever nearby (statue and lever are not there if you choose Family). Just make someone follow you in, then you step outside and pull the lever. Whoever followed you will be dead, replaced by your dog, alive again.