Simple answer is "You can't do it".
More complex answer is "You can, but you have to do things" (read below).
As already stated, you can get list of files via REST API. To make request to REST API you have to provide both public and secret keys. Widget does not support listing files as it uses only public key.
Of course you can make API request via AJAX, but you'll expose your secret key, and this will be on your conscience :).
What you should do is, build a backend view that fetches file list via REST, and makes it available to you frontend. Then you can add custom dialog tab that will show the list (somewhat similar is done in customization tutorial with "favorite files").