Ahh.. I understand. I thought you were saying you had your own app for the site.
Unfortunately, the Facebook App doesn't work that way. Apps have restrictions and they can't really cross communicate with other apps, like Safari. There's no way for Safari, and specifically your website, to know if the user has the app installed and if they are logged into it. I've never heard of the app working like that, but if you have an example of a site that does log you in based on the Facebook App or, even better, documentation from Facebook explaining how that would work, I'd gladly investigate further.
Thanks,
Alex