I do not think there is an easy way to disable this, but you have an option to delete it. If you go to the shared documents folder on the My Site, click modify settings and columns in the Actions list you have an option to delete the document library.
Instead of deleteing the Library you could remove the access to it, by going to the Shared Documents library, click modify settings and columns, chose change permissions for this library and remove all the site groups there, that would prevent anyone from doing searches and get hits from that library (since no one else than the "owner" will have permissions on the library).
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