In May, it filed a lawsuit on behalf of a handful of county magistrates in North Carolina, including Mr. Smoak, and argued that they should be exempted from performing same-sex marriages on religious grounds. The suit was withdrawn when the Republican-controlled Legislature overrode a veto by the state’s Republican governor, Pat McCrory, and enacted a law allowing them to refuse to aid such unions.

The group announced in July that it was representing Ms. Criner, the clerk in rural Irion County, Tex., who has also refused to issue marriage licenses to same-sex couples. Mathew D. Staver, founder and chairman of Liberty Counsel, said that no legal action had been filed thus far, and it is unclear whether gay couples have come forward in the rural county to request a marriage license. (Ms. Criner could not be reached for comment Wednesday.)

In Alabama, Liberty Counsel has for months backed some probate judges who balked at issuing marriage licenses to same-sex couples, and the group in July argued to the State Supreme Court that Alabama did not have to comply with the Obergefell decision. It noted that the Wisconsin Supreme Court refused to follow the high court’s 1857 Dred Scott decision.

As of mid-August, 11 Alabama counties were refusing to issue marriage licenses to all couples.

In an interview Wednesday, Mr. Staver, who is also a former dean of the law school at Liberty University, which was founded by Jerry Falwell, said that the Obergefell decision forced people “to accept and to promote same-sex marriage.”