Horses seem to have the combined health of their parents, so it would eventually add up to a horse with just about infinite health. In this picture, I bred a horse with 14 hearts and a horse with 12 hearts and got this 26 heart one. I'm not 100% sure if this is a bug or not though. It may be intentional, although I doubt it. Additionally, the health sometimes goes away with a relog.
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Duplicate of MC-14097.
There seems to be a lot of bugs with horses but I think it was intentional