Europe isn't financially situated, nor does it seem to have the social stomach for it.
We just don't spend it on an imperial posture and wars of aggression.
There are no European security interests in invading mid-eastern or latin-american countries, propping up dictators and toppling democratic governments in these regions and creating/financing the next wave of "terrorists", so that tab is all yours.
Furthermore, there is no "on your behalf" for the troops stationed in Europe. While some would argue that these are occupying forces in all but name, it's really in many cases just a convenient staging area for operations in the middle east.
In addition, and this tends to be forgotten, the nuclear shield is not a sign of generosity, but simply a tool to maintain US nuclear supremacy, by trading protection under the shield for a guarantee that those nuclear-capable countries under the shield don't, in fact, go nuclear.
US troops are stationed in those countries as assurance, human shields so to speak: if those countries are hit with nuclear strikes, US troops will die, ensuring a US retaliatory response.
Are you sure Europe couldn't manage its security by itself?
I mean, while keeping strong relations and cooperation directly with the US and inside NATO I don't understand why European countries wouldn't be able to defend themselves especially if we started thinking about a stronger cooperation between our defence systems or even a unique European army instead of keeping one for every country.