How to get and use the the number of days since the last comment?

To get the latest comment use get_comments().
get_comment_date returns the date of a comment in any format for PHP’s date().

Now it is easy. Let’s put the logic into a function to keep the global namespace clean:

/**
 * Returns the number of days since the latest comment.
 *
 * @return int
 */
function get_days_since_last_comment( $post_id = 0 )
{
    $args = array (
            'number' => 1,
            'status' => 'approve'
        );
    0 !== $post_id and $args['post_id'] = (int) $post_id;
    // Array of comment objects.
    $latest_comment = get_comments( $args );

    // No comments found.
    if ( ! $latest_comment )
    {
        return -1;
    }

    $comment_unix = get_comment_date( 'U', $latest_comment[0]->comment_ID );
    return round( ( time() - $comment_unix ) / 86400 );
}

Add the function to your plugin or to your theme’s functions.php.

To display a special message:

if ( get_days_since_last_comment() > 7 )
{
    print 'Looks like everything has been said.';
}

To get the days for a specific post (here ID 123) :

if ( get_days_since_last_comment( 123 ) > 7 )
{
    print 'Looks like everything has been said.';
}