Not according to the article.
The country’s parliament voted through the new law on same-sex marriage by a large majority, making it mandatory for all churches to conduct gay marriages.”
Churches in this article are congregations of the State Lutheran Church. That is the term meant by Denmark’s Churches. Denmark, unlike the United States, has a government-run Lutheran Church, that is sponsored by the taxpayers of Denmark, which is legislated by the government of Denmark. It is not synonymous with the Catholics, the Muslims, etc.