First, if such a transformation were to happen, it would imply, logically, that so far it hasn't (become a state), rendering that--vast--part of EU-scholarship which applies political-science-inspired models of quasi-statehood to the EU invalid. BTW, we know that the EU is not a state since it does not have what is considered the conditio sine qua non of statehood, at least since Max Weber: a monopoly over the means of coercion over a given territory.
What it does have, instead, is a creative, fast evolving system of distributed execution, utilizing a network principle (borrowed, incidentally, from the state of the art in the management of large corporations). Mahua Sarkar and I wrote about this in some detail in 2005, and the book also includes a similar analysis.
More interestingly, second, this raises the question of just what the point is. I mean, the point of the EU as a historical project in the longue durée, (see the last chapter in my recent book ;-)) seems to have been that, by inventing the EU, post-war west European states managed to find a reasonably creative solution to some of their deep structural problems of geopolitics. Those problems were caused, partly, by the loss of most of their colonial empires (causing a very significant drop in their control over global resources of all kinds) and a modern war universally perceived as very destructive. What was needed was a mechanism that would give them a field in which they can exercise a high degree of coordination, and eliminate armed conflict, among themselves--without forming a pan-European state.
Pan-European statehood has pretty much been an anathema for many powerful historical reasons, the biggest of course being that that was, pretty much exactly, the essence of the nazi Europa-Projekt, and that even given a largely democratic Germany, so far as that goes, the latter's overbearing influence within a pan-European state could easily unleash a complete nationalist reaction elsewhere. Not to mention of course the possibility that a future German leadership could have sinful thoughts and try to convert its initial position as the most powerful member state into a long-term structural advantage, ensured not only by the power of the German economy but also the by then pan-continental state.
Non-state-ness has worked reasonably well for the EU so far--especially if we consider that it is an absolutely new (i.e., completely un-tested) kind of geopolitical entity. For example, it has allowed the EU to engage a geopolitical strategy I have called the elasticity of weight: It allows the EU and its member states to appear in global geopolitical fora of all kinds either as a single entity (with a considerable weight if that is what is needed, working as an extremely effective cartel vis-a-vis small states dependent on trade with western Europe) or as a loose conglomerate of 27 states (with the appropriate number of votes in international organizations, for instance). A transition to conventional statehood would threaten that elastic strategy.
Full statehood would undeniably provide advantages a host of advantages to a number of European actors, of course. E.g., a common state would help reduce the much-discussed "democracy deficit" by making it possible for citizens of the European states to feel "closer" to Brussels (although I'm not entirely sure they would enjoy that closeness :-)). It would also make it possible to manage, and perhaps even avoid, crises of the Greece-Spain-Italy-Portugal kind, not to mention the fact that a federal state would provide a straightforward and matter-of-fact structure for the efficient and effective application of development resources, reducing regional and cross-state inequalities. Once that happened, it would also become possible to talk about a truly shared pan-EU social policy, etc.
And yet, because of the above historical legacies and the overwhelming power of the geopolitical considerations in the construction of the European project, my bet is that the EU will continue its current strategy of self-suspension in quasi-(supra-)statehood for as long as it can pull it off.
When that becomes impossible, then things will get really interesting in Europe--and the world at large.