Try this:

$q = new WP_Query;

$posts = $q->query( array(
    \'posts_per_page\' => 1,
    \'orderby\'        => \'post_modified\', // Sorts by the date modified.
    \'order\'          => \'DESC\',          // Sorts in descending order.
    \'no_found_rows\'  => true,
) );

// Note that $posts could have more than one post, if your site have sticky
// posts. However, the first post, unless filtered by plugins, would always
// be the most recently modified/updated one.
if ( ! empty( $posts ) ) {
    $post = $posts[0];
    setup_postdata( $post );
    ?>
        <p>Most recent site update: <?php the_modified_date( \'F d, Y\' ); ?>.</p>
    <?php
    wp_reset_postdata();
}