My Samsung Galaxy Nexus can't connect to unsecured web using wi-fi at home.
I have to edit the url with
https:// to load the webpage. Some pages work but most of them just won't load.
Your issue ain't related with Browser APP, it's a Network configuration regarding the proxy settings.
As you can read from your phone's user manual on page 63:
Proxy settings are used by Browser but may not be used by other apps.
Since you've mentioned that only "Browser" gets affected, and that all of them work fine under 3G, this leaves us with an WiFi settings issue:
Open the browser and give it a try. You should now be able to navigate without problems either with