I can't imagine such a thing. There could be a force comprising just EU citizens, but it couldn't be wholly independent. It would be dependent on US/NATO support to succeed. It would depend on Turkey to defend its border there. It would depend on both for intelligence. Who would the proposed "enemies" be? Russia? But Germany is building collaborative energy relations with them. Terrorism? Needs special forces. And so on....
I am in agreement with Christopher Nock . NATO is essential to provide protection and surveillance. I grew up in Canada and am an American citizen so I do not live in the EU. But an isolationist approach which having an independent army would possibly lead to, I do not think is a good idea. The cost would mean that resources would have to be reallocated. If you feel that NATO is not doing a good job, maybe a committee to review NATO actions and decisions would give you more control at a fraction of the price?
The existence of NATO is a historical fact. But it is worth considering who and what this organization saved from. In the 70 years since its inception, NATO has lead only one operation in Europe - against Serbia, which has never been part of the Eastern Bloc. The participation of European troops in operations in Iraq and Afghanistan is more reminiscent of colonial wars than “joint defense”. But even in this role, NATO did not officially help Great Britain in the war against Argentina, for example. In addition, the "border countries" at any result of the third world war will be completely destroyed, I can not understand why they need it? Russia's desire to capture them exists only in the inflamed brain of local politicians (nothing unites tighter as a common enemy, even invented). Russia simply does not have money for these gains (even assuming that there are intentions). I remember very well how happy my compatriots were when the "Eastern Bloc" collapsed, it is no secret that it existed at the expense of the USSR. Summing up, I can say that in any form, with the United States, without the United States, NATO does not solve any of the declared tasks, it only creates problems for the participants and a clear prospect of complete destruction in the event of a real war conflict, even "small".
I think nothing! If Europe feels strongly about going it alone, why not? It means they have already weighed the risks and benefits and the tilt is in their favour. It will be unthinkable that they could just wake up and think of bringing the whole of Europe down with a single policy.