We've helped other user's 'force' a login to Facebook on their site.. however, it's something that would have to be for any user's visiting your site. We could help you set it up so on only a specific page, the login pop-up will always come up, so it's not on every page. Again though, however, it's a pop-up, and not a modal, so users can ignore it and navigate around it.
The picture you show above is when a user logs into an app that is hosted within Facebook, that makes things easier. All the perms pop-ups in Facebook are part of the site, so they can be used to deny access to areas, whereas Facebook doesn't let us do the same thing on websites.. as they don't want to let admins force registration or granting of permissions.
Hope that makes sense, but let us know if you need more info,
Alex