The development reveals a shift in the balance of power from the government to the LDP, the reverse of what happened during the tenures of the prime minister's immediate predecessors.
The largest opposition party said it would stand for rights like supporting same-sex marriage and different surnames for couples, marking out differences with the conservative ruling LDP.
Comments by the vice finance minister quoted in a monthly magazine have inflamed ruling camp divisions, and seized attention, just weeks before the Lower House election.
The CDP is aiming to one-up Prime Minister Fumio Kishida, whose vows to reduce wealth disparity are in question after he walked back plans to review Japan's capital gains tax.