Here Microsoft is using their Desktop monopoly to boost their online search business and (this is the illegal part) restricting their monopoly product from using someone else's online search business.

That is incorrect. IE's search field is customizable; it defaults to Live Search but I can easily use Google, Yahoo, or whatever else I want. Google's issue is with Vista's desktop search through Explorer, which is currently tied to the indexing service built into Vista. It's easy to disable that service, but there's currently no way to tie the Explorer search box to another desktop search engine. Note that this has nothing to do with online search.