Starting with Mountain Lion, you can no longer restart Apache by unchecking and re-checking the Web Sharing option in System Preferences > Sharing. Mostly because it's not there.

To restart Apache, open the Terminal application (location in /Applications/Utilities/) and type the following command:

sudo apachectl -k graceful

It will prompt you for your password, but after you provide that, you're done (as long as there weren't any errors).