animal welfare

There is no way anyone can answer this legal question as we don't know what country you live in.

In theory, (at least PARTS of the US), you could take the person to ... say small claims court. In NYC, pets are considered property and are legally treated as such.

It is also pertinent as to where your dog was at the time he was hit.
Was he/she on your property, on your neighbors property, on public property?
I am very sorry for your dog. Is he/she still living?

Of course you can sue your neighbor, assuming you can afford the filing fee. Whether you have a valid cause of action is a different question. Good luck!

Apparently not.

Oh, I missed the "my dog died" part.

It is possible to sue him. Sounds like it is a claim you need to bring to your local small claims court. You aren't allowed to have lawyers in small claims court. You need to take pictures of the death. If there is nothing proving he did it, it is just going to be their word against yours.