There are a couple of things you can do. SEE an attorney. Preferably a real estate type that knows about the law there. Have him write a letter to this guy informing him and YOU about what the easment actually does, the rights of those concerned and the responsibilities involved with the gate. You NOW have a legal opinion and have informed this guy that you are ready to go that route. The letter and legal opinion will typically be around $100.
NOW, armed with the certified letter/opinon from the attorney let the guy leave the gate open. Catch him with a game camera, or better yet is the gate within 500 to 800 feet of your house? Put a motion detector alarm in the vicinity of the gate that will alert you when there is gate movement. Does he close it? Verify what he does with the camera or your eyes. Perhaps, an animal getting out after he leaves it open could even be arranged with damages and liability going to HIM.........Either way, that attorney letter does work at getting him to think about his behaviour because it IS going to cost him.
Some ol boys, because they have the "right" to do something will do that just get the reaction and to burn you. Nothing you can do about that emotionality; but you can protect yourself through the court.........God bless.......Dennis