There isn't protection in The Privileges Clause or Due Process Clause that protects citizens regarding their choice in sexuality as you suggest
The privilege clause was instituted in 1866 after slavery ended, and it was to protect freed slaves in their right to property ownership, etc (Privilege)
Privilege Clause: No State shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States;
The due process is to protect citizens from being deprived of life, liberty, property, as defined below, and against unequal protection of the law, such as inability to afford an attorney, that is why public defenders are appointed to the poor and destitute to satisfy the equal protection clause