Agree with Rebecca, they certainly grieve, although it's depends a lot on the horse how bad it is. If possible, it's always best to give let the horse see his buddy after death. He may still act sad/withdrawn for a while, but they really do understand death when they see it... I've known horses to freak out/panic for days if the buddy was just taken away and put down, but never if they get the time to see/smell/process. (I worked at a retirement farm for years, saw it happen many times)
They usually like to be away from the herd and be were they always hung out he might not eat or drink water and just basically look like a human that is greaves get from a loss just not there they will be thinking a lot
Horses can grieve and I know from personal experience with a gelding of mine that they can become secluded but with some time he came around and slowly returned to his old self. It also helps if there is another horse or animal for them to be around.
They are just like people, they get depressed and don't act the same. It's bad for them so if their companion died you might need to get another. Like a goat or a pony.
I also think they greive, A friend has a mare that that was really nice and rideable saw her baby killed by humans due to a illness, and ever since has hated and cannot be riden to humans
I agree with Rebecca ad I think so too! Of course we'd like to be more similar than us, but they also have feelings and I think they miss their companions etc.
It is my opinion that horses can grieve. I further believe that grieving takes place, for the most part, when the animals can see the dead or dying. Once they are apart, grieving normally stops very quickly. Many animals grieve for days with the loss of an offspring, I believe horses fall into this category.