Wordpress Customizer inline css to css file

You can use file_get_contents for getting all css written in the customizer.css file and file_put_contents for adding new css rules into customizer.css file.

Your code will be:

function customizerCSS() {
    $color = fw_get_db_customizer_option('body_background');

    $custom_css = '
                html, body {
                        background-color : ' . $color . ';
                }';

    $file = file_get_contents(TEMPLATEPATH.'/customizer.css');

    if (strpos($file, $custom_css) === false) { // for not rewriting file every time, when there is no changes
        $file .= $custom_css;
        file_put_contents(TEMPLATEPATH.'/customizer.css', $file);
    }

    wp_enqueue_style('custom-style', get_template_directory_uri() . '/customizer.css');
}
add_action( 'wp_enqueue_scripts', 'customizerCSS' );

We are getting all new css rules from $custom_css, everything from customizer.css and comparing them. If there is new css rules in the $custom_css which aren't exists in the customizer.css stylesheet, then we just writе them into file.

Edit:

Function below will override customizer.css file with values from $custom_css variable every time:

function customizerCSS() {
    $color = fw_get_db_customizer_option('body_background');

    $custom_css = '
                html, body {
                        background-color : ' . $color . ';
                }';

    file_put_contents(TEMPLATEPATH.'/customizer.css', $custom_css);

    wp_enqueue_style('custom-style', get_template_directory_uri() . '/customizer.css');
}
add_action( 'wp_enqueue_scripts', 'customizerCSS' );