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:
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