The EU will slowly transform itself into a federal Government, essentially turning the EU into one country will all the members as states, much like the USA.
When this happens, one of two scenarios will take place:
1. Other nations around the world will start forming their own unions to help them compete internationally along side the EU and USA. There are already things like the African Union, but they're nowhere near as "far" as the EU. When the world has several large unions, they will start cutting deals, breaking down the barriers between them.
2. Once the EU has fully developed in Europe, it will open its boarders, and slowly start letting members in from Asia, and then Africa, then the Pacific, and finally, the Americas. This process, IMO, would be much more successful than option A for uniting the world, but it would take far longer to achieve (far, FAR longer).







