The European missile defense system is designed to destroy any ICBMs launched from Russia. Although Russia isn't the Soviet threat it used to be, the country is a wild card on the world stage as it sides with China on several issues and has backed Iran. If Europe gets a missile defense system, Russia will build up some kind of military hardware on its side of the border as well.
European countries would consider missile defense systems in other countries a threat to their own security. They might counteract this threat by ramping up their own defense systems. The NATO defense systems are already considered a threat by the Soviets and they are moving their defenses closer to the EU in response.