I think the biggest issue is one of "perspective".
OP is upset because "something" is being left on "his" road.
We're talking about a few horses, maybe not even a few horses every day. More likely it is a couple horses during the week, and a few horses at a time on weekends. I seriously doubt the riding stables operation is very big. They probably use "his road" to access other trails because it is SAFER for the new and untrained riders to ride down a seldom used dirt road then to have to ride around on what would likely be a longer and busier traffic road.
I'm sure they're not doing it just to be JERKS to Tom.
We're also talking about horse apples, which are biodegradable (and do so rather quickly). And horses are not continuous "apple droppers". They can go for miles on a trail without dropping a single apple. So unless Genghis Khan is staging a counter attack and using "Tom's Road" to move his rebel hordes, there just can't be that many apple piles. Are these new, untrained and (presumably) young riders riding all across "Your Road", or are they riding mostly single file down the side? I very seriously doubt this is dump trucks full of manure here. More than likely it is a few apple piles spaced rather far apart, likely down the side of a rural dirt road.
I think Tom is too wrapped up in the "IT'S MINE!" mind set and is unwilling to look at this objectively. I seriously doubt he has looked into the road situation from the standpoint of who is actually legally allowed to use the road, and for what purposes. If "His Road" connects to other roads or trails, he might be "horse apples out of luck", and there was likely an agreement with the county or township that gives others the right to use "His Road". And the impression given from his posts, this is a somewhat rural DIRT road that connects to riding trails.
Also might want to look into local ordinances about livestock moving or use of the road. I've never seen a rural road used for livestock movement that hasn't had "apples" on it. I very seriously doubt there's anything in your local ordinances that prohibit it.
This could end up blowing up in your face in ways you haven't thought about. The Stables could turn around and sue you for harassment.
I suggest you take a breath Tom, and gain some perspective.