Can a veterinarian write doctor before his name?

It depends on the custom of the country and the professional veterinary society. In the United States, veterinarians are granted the degree Doctor of Veterinary Medicine and are accorded the title of "Doctor" upon graduation.

It depends on which country they work in and which country they obtained their veterinary qualifications in. In Australia if you are a registered veterinarian and don't have a PhD you can apply to the registration board of the state you work in to use Dr as a courtesy title.