It says the following (I edited the paths and domains for security/privacy):
2013/06/02 13:22:48 [error] 9861#0: *6 directory index of "/home/path/whatever/" is forbidden, client: 220.127.116.11, server: _, request: "GET / HTTP/1.1", host: "www.domain.com"
2013/06/02 13:22:49 [error] 9861#0: *6 open() "/home/path/whatever/favicon.ico" failed (2: No such file or directory), client: 18.104.22.168, server: _, request: "GET /favicon.ico HTTP/1.1", host: "www.domain.com"
I thought that was the correct path, but maybe that's the problem. You talked about "/var/www/default/public_html", but I don't have a that directory (there is no "www" in "var").
My files are in "/home/path/whatever/domain1", "/home/path/whatever/domain2"... so I guess "/home/path/whatever/" is the "directory that holds the files that should be served in case nginx receives a request for a domain name it can't recognize". Please correct me if I didn't understood it correctly.
Maybe it's also relevant to mention that I'm using Drupal and my nginx config is the one described in this page: http://wiki.nginx.org/Drupal (it's the first result that appears when you search "drupal nginx" in google, so I guess it's a standard and well tested config)