There isn't much the EU can do about it really. If a US company doesn't want to charge the destination countries sales tax they can still send the goods / services and the customs in the receiving country will add import duties and sales tax to it.
Fact is, the problem is just as bad the other way around.
When doing minor transactions like via PayPal anyone in the EU needs to add sales tax to the transaction. Unless they can really prove and I mean "really" prove that the recipient is located outside the EU. And in case recipient is within the EU and the seller doesn't want to add the sales tax he'll have to have proof of the recipients local tax id and subsequently fill out a tideous form. But that still leaves room for scrutiny because how on earth will a $40 transaction bear to first have to check the validity of a tax id in a foreign country and then have to fill out a form. How's that for beaureaucriciy?