The horse is out of the barnIf the owner knowingly let the horse out into the roadway, then it's possible he will be held responsible for it crossing your path and being hit. If the owner were unaware that the horse was set loose, such as someone vandalized his fence and it escaped, then the owner may not be found liable.
Also, in some western states, there are still open or free range laws that allow domestic and farm animals to roam free. These laws are older, so some have been updated to take into account more modern times and at least require for livestock owners to keep their animals from roaming unattended on the right-of-way on highways.For example, Texas Agriculture Code section 143.102 basically tells owners to keep farm animals off Texas highways and then section 143.103 gives immunity to someone whose vehicle strikes, kills, injures or damages an unattended animal that is running at large on a highway and therefore is not liable for the animal except if the driver is found to be grossly negligent in the operation of their vehicle or showed willful intent to harm the animal. If, however, your state's laws allow certain animals to roam free and puts the burden on the driver to keep from hitting these animals, blame could actually fall on you for injury to the horse.