XNA Framework users won't be given the luxury of freedom, but rather an easier to use API for data transfering across an already established network connection.
Think of it as something similar to DBP Multiplayer, so no way to physically alter the connection type or requests only the data being sent.
Microsoft requires every game to be Live Aware, and those with multiplayer (over a network) options are using Live and the Networking API. So you don't have to mess with the background information and such just need to concentrate on the actual data being sent between.
There really wouldn't be any way to make a browser for the 360 with the XNA Framework (let alone XNA Game Studio).
I very much doubt there will be an internet browser on the 360, there just isn't any need for it.