The court dismissed the plea on the grounds that under the law of land it is not entitled to change legislation, Echo Moskvy radio reported.
Last month two Russian males had applied for a marriage license from a Moscow registration office. One of the men - Eduard Murzin is an MP of Russia's internal republic of Bashkiria.
Murzin, who is a heterosexual, claimed that he wanted to promote human rights
He and his gay "partner", Eduard Mishin, chief editor of a gay web-site, had widely publicised their plans to register their marriage in an attempt to draw attention to the movement for gay rights in Russia.
But the registration office has refused to issue a marriage certificate, because same-sex marriages are unlawful.
Murzin believes the Family Code contravenes the Russian Constitution and the European Human Rights Convention and said he would challenge the Supreme Court's ruling in the International Human Rights Court in Strasbourg.