No, the European Defense force is not a feasible notion, because it would be too hard for the nations involved to reach a consensus. The European Union does most of its action by consensus. If every nation in the European Union had to agree, they would never actually make a decision to take military action. The force would be very inefficient.
Unfortunately, an EU defense force isn't a feasible option at the moment, especially not one comprised of every EU nation. These nations are so far apart economically that only a handful of countries would dominate the decisions. Plus, the larger countries would be expected to defend the smaller ones, which causes too many problems rather than solutions today.