Marriage is not and never has been a “right”. If it were a “right” the state could not tell you that you can’t marry your sister/brother. It could not tell you that you can’t marry one of your children. You must get permission from the state to marry.
Also, if it were a right, you could “demand” a partner.
Yet the "rights" line of reasoning was argued in Loving v. Virginia (1967) and other marriage cases that followed it. Chief Justice Earl Warren's opinion for the unanimous court opined that:
"Marriage is one of the 'basic civil rights of man,' fundamental to our very existence and survival.... To deny this fundamental freedom on so unsupportable a basis as the racial classifications embodied in these statutes, classifications so directly subversive of the principle of equality at the heart of the Fourteenth Amendment, is surely to deprive all the State's citizens of liberty without due process of law. The Fourteenth Amendment requires that the freedom of choice to marry not be restricted by invidious racial discrimination. Under our Constitution, the freedom to marry, or not marry, a person of another race resides with the individual and cannot be infringed by the State."