Since CERN is exactly on the border between Switzerland and France, the exact country where web was born can only be determined by considering some pretty funny facts.
David Galbraith updated his post on where the web was invented to include the juicy tidbit that the building in which TBL invented the web is in France, not Switzerland.
The CERN is foremost an international place, so I think that no single country can claim the right on the origin of the web. Tim Berners-Lee is Brittish, Robert Cailliau is Belgian, and the first web server ran on NeXT computer that was a brainchild of Steve Jobs. If a single group of people can claim to gave birth to the web, that are scientists.