First of all, root
directive inside location is perfectly valid, but there are few cases when it's appropriate and this one is definitely not the case. Read common pitfalls.
Second, relative root path (root html;
in your case) is relative to nginx default path. Do you know what it is? Yes, it's /usr/local/Cellar/nginx/1.4.6
as stated in your post. So root html;
equivalent root /usr/local/Cellar/nginx/1.4.6/html;
. I guess, there is no PHP file.
Third, nginx does not execute anything. In this case nginx proxies your request to fastcgi application on port 9000 (I guest, it's php-fpm or something like this).
Also, there are lot of useful messages in error log. Try to look into it.