Do Property Owners really own their own land?

I am baffled about a decision that was made by the Board of Directors of our HOA.  We live on a golf course and would like to have a private backyard, where golfers just don't drive or walkon our yard.  We love the view and we do not mind the golfers, we just want to sit in our backyard without having to say Hi to everyone that passes by.  We also want to have a dog and we do not want him chasing balls, that would not be fun. We applied for the fence and the board approved it, but we had to get approval from the neighbors.  The neighbor to the left signed and the neighbor to the right, whose home is way above ours, did not sign and here is why.  He hates dogs and he thinks if he will not let us do 15 ft of fence on his side, it will prevent us from having a dog. Now, we will have an L-shaped fence with a 15ft. area open, so we asked to plant a hedge and again he said no, then we decided to do an invisible fence and he again said no. We even told the board that we would put the fence away from the propery line, no again.  We can not even move potted trees in that area, unless  he agrees. We will have an opening that will allow anyone to walk through and a dog to get out, which we do not want, I suppose we could get a pool and then we would have to have a fence, but a dog is much cheaper. We tried to sell our home, but buyers would like a more private backyard, that is another reason we would like to do it.  Everyone around us has private backyards.  Does anyone have any ideas, suggestions or know something that could help? Maybe I could put a sign in his yard, "For Sale by Neighbor"