The problem is that those in charge in the EU will sit on their collective hands and do absolutely nothing until any problem has become so massive that they are dependent on US military assistance in order to hope to survive.
The problem is that many EU countries are dependent on Russian oil and gas. US military can't solve that problem.
Of course, Russia is also dependent on EU buying said resources. A disruption of trade would, at the very least, topple Putin. He's counting on EU being unwilling to take the pain, but it's becoming increasingly clear he's trying to rebuild Soviet Empire. Sooner or later he'll miscalculate, and at that point it's a question whether he can gets removed from power before starting another (world?) war as a desperation move.
But no, Russia cannot win a war with the EU. EU has 3.5 times as much population and almost 8 times as large economy. Even with all the inefficiency inherent in coordinating multiple independent militaries, it could only end in Russian defeat or nuclear escalation.