The European Union is powerful in itself. But it also has a contender which is the United States and other countries. I think that it is not a question of the European Union becoming the world's policeman or the United States being the world's policeman, it is an issue of the fact that there needs to be a world's policeman no matter how big or small. If the European Union can do more than what the United States and other countries have done, then yes, the E.U. can become the word's policeman.
Citizens of the world have evolved at different paces. Behaviour that was acceptable in Europe 50 years ago is often now a crime . Yet this behaviour is still normal in other countries . Europe needs to lead by example not policing . Children in Europe are being forcibly taken from their families and legally adopted ! For countries were families still are the heart of their societies this abhorrent .
E.U should not and could not be the world's policeman. It will not only create conflicts around the world which will never be resolved but also crumble the economy. No great empire has ever lasted once it got as big as controlling the whole world. It will simply not work and is asking for dangerous results.
There is no world police scenario that has worked out and there's nothing to indicate the EU is about to throw their hat into the ring trying to be it. World police over extend their influence, make much of the world hate them and don't accomplish their goals. It's a concept that has an unblemished track record of failure and I doubt the EU is interested in giving it a go.
No, why would the E.U become the world's policeman? I don't see any benefit in doing this. Certainly they can try but I don't see it being in any way successful. History sure does always repeat itself. What I mean is that by being the world's policemen, you are asking for conflicts and possibly world war.