A traditional-values law firm announced Monday that it has filed a lawsuit to overturn New York state’s new same-sex marriage law, claiming the state used a “corrupt legislative process” on its way to becoming the sixth and largest state in the country to sanction gay marriage.

The lawsuit, which the firm said was officially filed Friday, claims violations of New York State Open Meetings laws, irregularities in the Senate voting procedures, and “unprecedented Senate lock-outs” in which lobbyists and the public were denied access to lawmakers.

Gay marriage was enacted soon after the Senate voted, 33-29, in favor of it on June 24. The law went into effect July 24...