That isn't the case; Denmark simply does not charge for acute emergency visits, and does not send a bill. In addition, they pay for their own residents who incur such bills within the EU on trips under 30 days [1], which is part of the controversy: a French person who ends up in a Danish ER is not billed, but a Danish person who ends up in a French ER is billed, with the bill paid by the Danish government. Most Danes would prefer that each country provide free emergency care to all, like they do with fire and police.

[1] http://www.sos.dk/en/Corporate/OurNetwork/ForMedicalFaciliti...