I have requested that my entire site is removed from the index using
the URL removal tool. My site is not intended to be publicly
accessible, but an error in a .htaccess file had left parts of it
open. This error has now been corrected so that the entire site
returns 403 forbidden errors.

I see that the URL removal tool says you must return a 404 or 410,
block using robots.txt or block using meta noindex.

Will returning a 403 work too? I really don't want to re-open the site
to the world.

My URL removal request has been denied, so I guess my question changes
to: Is there any way to remove my site from the index without opening
machine access to the public?

In order to remove your site you'll have to meet the requirement of
having a robots.txt file that blocks your entire site.

The robots.txt file's content should contain the following two lines:
User-agent: *
Disallow: /

Here's an article from the Webmaster Help Center for more info on
setting up your robots.txt file to block your entire site:
http://www.google.com/support/webmasters/bin/answer.py?answer=35302