That sounds incredible! You just gave me a new dream ;)
But personally I don't necessarily think it would be a good idea to buy a horse for this. Maybe if you settle somewhere and buy a horse, you could later plan a ride like this, but I don't think it makes sense to land here, buy a horse and take off.
But like others said, what about traveling slowly and riding a lot? You could do a 50-state trail-riding journey, going on 2-5 hour trail rides in every state. It would be a lot less risky for you and the horse, and if you are serious about owning a horse in the long term, that would be an incredible way to figure out exactly what kind of horse you want.
"...and may your life be filled with good horses." ó Buck Brannaman