What does "put the imprimatur of the State itself on an exclusion" mean in the following context?

Many who deem same-sex marriage to be wrong reach that conclusion based on decent and honorable religious or philosophical premises, and neither they nor their beliefs are disparaged here. But when that sincere, personal opposition becomes enacted law and public policy, the necessary consequence is to put the imprimatur of the State itself on an exclusion that soon demeans or stigmatizes those whose own liberty is then denied.

Does "put the imprimatur of the State itself on an exclusion" mean "the State approves of excluding the gay marriage" or "exclude the State from approving of gay marriage"? What confuses me specifically is the definition of "exclusion" here.

Originally, an imprimatur (Latin for "let it be printed") was official permission from the Catholic Church to print a book.

Since then, the word has come to mean official approval generally. In this case, the writer is saying that for the government to forbid same-sex marriage is to (effectively) give official approval to homophobia.