Yes you can.
When a browser accesses the server without authenticating it accesses it via the 'anonymous' account. /Public is the default home folder of the anonymous account, so that is what will be presented to it.
The home folder can be changed and it can be different for different sites, so you can configure the home folder of anonymous to be, say, /Public for the Public site and /Internal for the Internal site. The owner of both these directories should be 'anonymous'. You can configure /Public to point to one Windows directory (e.g. C:\Webroot\Public) and /Internal to point to another (e.g. C:\Webroot\Internal).