Display users uploaded files as posts

If I am understanding the question correctly, you should be able to do a query for the media by user id. Something like the follow (untested) code:

//only do this if there is a user logged in
if( is_user_logged_in() ) {

    //get the currenly logged in user
    $user_ID = get_current_user_id();

    $upload_args = array(
        'post_type' => 'attachment',
        'author' => $user_ID
    );

    $uploads = new WP_Query( $upload_args );
}else{
    //what to do if there is no user logged in.
}

Hope it helps!