Afaik, Denmark doesn't require the EHIC for acute emergency care. Anyone is eligible for unbilled emergency care, with some exceptions to combat medical tourism (if you're found to have traveled to Denmark for the specific purpose of entering an ER to treat a health condition). The EHIC qualifies you for some additional care, e.g. childbirth. This is actually a big current sore point in Danish politics, because many other EU countries don't extend a reciprocal courtesy.