Windows – Web service listen on Public and Private IP address

The short answer is no, unless the application specifically supports it. Otherwise it's up to the IP stack to determine how things get sent out so doing something like putting a weight on one interface is what you have to do.

A workaround for a browser would be to install a local proxy. WebScarab will allow you to do this, if you don't mind installing it on a server. You can set up a local proxy bound to a specific IP and then set your browser to use the local proxy. Thus the request will be sent to the remote server using the IP that the proxy is bound to.

If you use an internal proxy follow this guide to chain WebScarab to that proxy.

The only thing is that you're not saying "all addresses should have access" -- that's done in your firewall(s) and/or the server software and/or other security layers like tcpwrappers.

0.0.0.0, in this context, means "all IP addresses on the local machine" (in fact probably, "all IPv4 addresses on the local machine"). So, if your webserver machine has two IP addresses, 192.168.1.1 and 10.1.2.1, and you allow a webserver daemon like apache to listen on 0.0.0.0, it will be reachable at both of those IP addresses. But only to what can contact those IP addresses and the web port(s).

Note that, in a different context (routing) 0.0.0.0 usually means the default route (the route to "the rest of" the internet, aside from routes in your local network etc.).