If someone adamantly refuses to listen to advice from people who are older and wiser, then there may not be much you can do about it -- except to pray.
I'm reminded of the people of Jeremiah's day, who refused to listen to his repeated attempts to get them to turn back to God. They told him, "It's no use. We will continue with our own plans; each of us will follow the stubbornness of his evil heart" (Jeremiah 18:12). But Jeremiah kept on preaching -- and as you and others in your family have opportunity, you should keep on expressing your concern.
At the same time, let your cousin know that even if you don't agree with his decision, you still love him, and you want what's best for him. After all, his marriage may turn out better than anyone expects, and if so, he needs to know he's still part of your family. And if it doesn't last, he'll need your love and encouragement. The prodigal son left home against his father's will, but the father still loved him, and welcomed him back when his life fell apart (see Luke 15:11-32).
The most important thing you can do, however, is to pray for your cousin, and encourage him to seek God's will for his life. Right now, he isn't thinking about God, but God hasn't abandoned him, and his greatest need is to put Christ first in his life. This is true for you, as well.