Could moving my WordPress installation from /var/www/html/ to /var/www/ break the application?

You’ll have to make sure that your web server knows about the change, but it shouldn’t break anything inside WordPress. All the WP links are based on its URL, not its location on a disk. There are WP functions that reference disk locations (eg plugin_dir_path(), but they should all be relative to the site’s root directory, not /.