Can you just use the domain to house your database and code? Do you have to have a "real" front end webpage?
I think what you're asking for is "do you need a complete website with index page and links, or can you just host game.html". If I'm right the answer is: no you don't need a real website. Aslong as facebook can access the game, and it is playable you're fine.
I would not recommend it though, what if a player goes straight to that link in his browser (www.website.com/game.html) but isn't online on facebook, does that affect how the game plays?
Useable for minigames that don't need usernames though.