Vets can be found in every single town other than Leaf Isle. The buildings appear to be small, white buildings that have a sign on it: It says VETETARIAN.
Once you click a vetetarian, you will have two options: LOOK and CURE(or whatever it is, I forgot). Once you click CURE, it'll bring you to a screen that has a list of injured horses. On Life Cycle, there will be a gelding option as well. Note that they charge a fee for curing your horse, the fee is usually arround 6000-7000, but bring at least 8000. Veterinarians in different towns charge differing rates for their services, but all of them can heal your horse.
Overall health will decline steadily the longer an injured horse remains without veterinary care. This decline continues to increase even when you are offline, so it is very important to take an injured horse in for veterinary care immediately after an injury.
Note bene: a Veterinarian can't cure hunger and thirst, proper care is still required.