I was extremely ironic Apenis lol .. but I now it will make your day so what will happen if I leave a boiled whole salmon on my counter ?
Dry it with a hair dryer and put a decent amount of salt on it. Keep the hair Dryer on it if you notice any moisture. Also, store the fish as high up as you can (heat rises). If you REALLY want to preseve it, just salt it and smoke it.
Now if you just want to set it on your counter, you are going to have to eliminate as much moisture as you can in your house and keep it dry as you can. You will also need to keep bacteria at bay which is not too difficult to do if you can keep it dry. It can be done if you are so inclined to do so. Also, being that you boiled it, you effectively killed any surface bacteria and you also partially dehydrated the fish by converting any water in it, to steam. This will make it a bit easier.
Also, have you EVER heard of Smoked Salmon or Salted Fish? Its been done already with Salmon in case you didn`t know.