Can you set the directory from which the Media module populates its Library tab?
The Media Browser does not work with a physical path, but with the file managed table, and Drupal file system uses the uri of the file to get to the actual file.

Go to the views settings page admin/structure/views/settings and turn on 'Show the SQL query'.

Then open up the media browser and look at the SQL query. It starts of with 

SELECT file_managed.timestamp AS file_managed_timestamp, file_managed.fid AS fid, SUM(file_usage.count) AS file_usage_count
FROM 
{file_managed} file_managed


which means it is using the file_managed table as its base. That comes from the file module in Core. Looking at that table definition in PHPMyAdmin for instance will reveal a fid as the primary identifier, a filename, a uri, filemime and some other fields. The uri is the important part as it tells Drupal how to get to files. The uri is prefixed by the stream wrapper (PHP classes that provide a common interface for getting to resources) and the two common Drupal ones are public:// and private://.

The media module already provides a remote internet sources wrapper, and Remote stream wrapper module provides a Remote Url plugin for the media module.

You can also write your own. Writing stream wrappers