No, the EU is an effort by the European members to marginalize US economic power. NATO is obsolete these days and is a one sided , as far as benefits, european entity.
If EU members feel threatened by another sovern state, THEY need to form their own fighting force, fund it, field it , and command it.
NATO is obviously relevant today, as Russia is on the move and non NATO nations are looking to join the defense alliance for protection and unity from Russian conquest.