and you have cleared him of UTI's or other medical problems, then you have to remember that peeing outside is his way of marking his territory. Can he possibly see other outside cats from this location he has chosen? Even if they are outside, he will consider them a threat and spray his marking to warn them to stay away. Perhaps you can block his view of this world temp. until he stops doing it. Also, once he has marked a spot he will return to it time and time again because he is following the scent he left, and when it starts to recede, he will want to "reinforce" it again. Use a good strong cleaner like Simple Solution, or call your vet so that he can recommend a good cleaner that clears up ALL the odor. Cat pee is very strong, and the carpet is probably saturated with it at this point. You can also try throwing a good thick throw rug over the spot, spray the throw rug with a good citrus scented room deodarizer as well.
He is a cat, and they have specific rules of behaviour handed down through the generations of his wild ancestors. You can not always make them conform to what you want them to do, you have to think like them and try to figure out what is causing this. As I do rescue work, I see all the time the after-effects of cats whose people have given up on them because of behaviour issues. Try all the suggestions that have been offered, do an internet search for more answers if you need to, and then, I guess, if it doesn't work to your satisfaction then take him back. But it could be just one minor change in his enviornment that is setting him off, you just have to try and find out what that is.