Well they will unless we cede control of the root DNS servers. Different discussion, but the internet is a network of networks. It is decentralized, but there are underpinnings that only the U.S. controls, and I strongly hope we always will.
That said. In this case I trust our FCC a lot more than depending on a foreign based company to provide a service that is free and open (for me that is saying a lot, but it is true). In fact if a "global" service like this happens I would expect it to be provided with a great deal of filtering and blocking to limit dissident content.