People are entitled to believe as they feel is right. Believing that God ordained marriage to be between a man and a woman is no more changeable than your sexual preference is. And it is possible for your cousin to believe as he does and love you.

You have the right to choose who you talk to and spend time with. If you choose to spend time with your family or others whose beliefs you do not like, you may need to make certain topics off limit for conversation. They don't need to harass you over your lifestyle choices, and you don't need to insist that they change who they are or what they believe either. They have rights also.