This page explains the methods for healing damage that your horse has taken.
Board Your Horse
Whenever you board your horse at a stable, your horse's health is restored to full.
Whenever you use the Sheikah Slate to teleport to a Shrine or Tower, or when you use the Travel Medallion (available if you purchased the Breath of the Wild Expansion Pass), the registered horse that is currently out of the stable will have its health restored to full.
You can revive registered horses that have died. Southeast of Highland Stable (near Ka'o Makagh Shrine), you will find Malayna Spring at the end of the road. If you examine the spring and pay 1000 rupees, Malanya will appear, and you can ask to revive any horses that died, if you had registered that horse at a stable prior to its demise. The first horse is revived for free, but each subsequent horse costs one Endura Carrot to revive.