But... but... what about childish power fantasies and world conquest?
In all seriousness though, the EU does not currently extend to defense; it is currently only a political and economic union. France and Germany have proposed extending the EU to include common defense, but as of today it had not yet been ratified by either the European Council or the member states. See, e.g., the Treaty of Lisbon. (Note that the Treaty of Lisbon technically removes the unanimity requirement in place of a "double majority" threshold (meaning >55% of vote from >65% of countries) beginning in 2014.)
Fun factoid: France lets Switzerland use part of its territory to defend Geneva. And also let it fly its plane over FR territory because CH is so small that it's not practical for pilots training.
M.A.D. still exists by the way. It's still why all major nations are at peace and why the EU gave so much flak to the US a few years ago when Bush wanted to install a nuclear shield. He later had to back down or serious stuff would've happened. (afaik)
Also, according to wikipedia, Europe has 150 to 240 nuclear warheads. Plenty enough to destroy all US major cities.