LGBT-identifying Americans have long been discriminated against, and the legal marriage system in America is biased against them. By creating an institution of marriage and allowing LGBT-couples to enter into said institution, it will ring true to the Fourteenth Amendment.
Within the Fourteenth Amendment, the Due Process clause restricts the government from interfering with a person's life, liberty, or property. Liberty is allowing for an American to marry whoever they so choose and creating a life between then, and owning property and investing in their community.
Failure to grant same-sex couples these same rights is a Constitutional violation.