It's a weird document. It goes to great, very great lengths to emphasise the success of the private sector in building and maintaining the backbone of the internet, whether we're talking early days or right now. Basically every management aspect of the internet (including here in Europe) is handled by private companies, relatively to completely free from government meddling.
The document dedicates an entire section to the success of ICANN, and only points to the lack of international oversight as a flaw. What this basically means is that the EU admits that the current setup works just fine - even without its input. If it's all been such a success, why change it?
The biggest reason the EC brings forward is local accountability. The internet has become such a vital piece of everyday life that if something were to go wrong, like interruptions in service or other forms of breakage, citizens will hold their local governments accountable - they won't have the technical knowledge to place the blame where it could (possibly, of course) belong: ICANN. ICANN's influence stretches far beyond the United States border, and if something goes wrong, the EU and its member states have no way to respond.
There are two sides to this issue: a practical one, and a philosophical one. Practically, there is absolutely no reason whatsoever to change all that much about how the internet is governed. ICANN and the US government have ensured that the internet has a free flow of information - no matter how objectionable (but still legal) some content may be. This is intrinsically a good thing, as that's what the internet is all about.
The philosophical side of the coin is that it is an absurd idea that the US government - a government most of the world did not elect - has such a possible influence over the workings of the internet, while the rest of the world sits by the sidelines without any form of influence.
When it comes to government influence, I believe in a case-by-case approach. Stating that government control is always bad is just as short-sighted as stating that the government should control everything. The ideal solution is somewhere in the middle, where individual aspects of society are evaluated to see if they might benefit from some governmental handywork.
Drawing this closer to the internet, I see no reason to extend the influence of any government over the internet. It works fine the way it is, and the US government and ICANN have done an outstanding job. As much as I personally dislike the US governmental system, there is no denying that the internet is a success story.
On top of that, if there is one institution I really don't want even an inch closer to influencing the internet, it's the EU. I don't trust the EU one bit when it comes to this matter, as they have shown an almost uncanny ability to regulate even the tiniest of details of every day life.