I think it would be very difficult to create one big army in Europe for a number of reasons:
1: Budget, the amount of money every country spends is extremely different, depending on what kind of defense or military the country operates and what the purpose of that military is.
2. Types of Weapons. In the US they all use basically the same types of weapons varying from handheld to automotive or aircraft mounted. In europe every country uses different types of weapons all depending upon the climate and usage for that specific country. We also have a huge amount of military hardware manufacturers, Sweden, Finland, Germany, France, Spain, Italy, Uk etc all build weapons adapted for a specific purpose...
3. Climate. Just think of the different climates that we have in europe, from almost desert like areas in the south to extreme winters in the north, these all call for different tactics, different weapons and knowledge. How is a general from spain to know how to fight a war in norway...
I think Europe will have a cooperation of military forces but never a "one single armed force".