The news reports on this are particularly bad.
HuffPo is simply saying that gay couples can now legally marry in a church. On the other end of the bad reporting spectrum, the
Telegraph is reporting that gay couples have the right to marry in any church of their choice. Neither story is accurate or complete. The reality as far as I have been able to discern is that gay couples can demand to be married in a church starting on June 15th.. But the new law only applies to the official Lutheran Church which is effectively a department of the government. Individual clergy can refuse, but the bishop must find a replacement. It is worth noting that the minister of religion in the Danish government is a self described agnostic.
HT: Dr. Tighe