Nathan,
No, you can't do that. It'd be nice, but could also get unorganized quickly as Facebook likes discrete areas for things. If a comment is about a specific page/URL, then it should be posted to that page. If you have a long thread on your Facebook Page that's really about a URL on your site, it can be awkward and the conversation could easily go astray with other postings.
I can't always speak for Facebook's intentions, but I know how they work after dealing with them for many years, and they really push the 'branding' of a Facebook Page. If you mix that too much with a live site, then Facebook can become secondary.
Hope that all makes sense and answers your question,
Alex