That used to be possible up to and including API v3.2, as described under https://developers.facebook.com/docs/plugins/faqs#faq_1603507626630008
The comments for every URL can be accessed via the Graph API. Simply make an HTTP GET request to:
https://graph.facebook.com/v2.6/?fields=og_object{comments}&id=<YOUR_URL>&access_token=<YOUR_TOKEN>
That doesn’t appear to work in v3.3 any more though, only results in an error message saying,
(#100) Tried accessing nonexisting field (comments) on node type (OpenGraphObject)
I can’t find an explicit mention of this in the changelog right now though.
And even in API versions prior to v3.3, it seems you can only get the created time and message (plus maybe a couple of other meta fields), but specifically from
– info about the actual user making the comment – is not available any more.