My cousin does not know that I know he is gay,

Ok, long story short. I am a straight man but do not have anything against anyone being gay. I encourage everyone to be themselves, not matter what.Little backstory: I have this cousin (20 year old male, lets call him Joe), whose father passed away about 10 months ago and his mother is a drug addict so he wants nothing do with her. He's lived with us for months at this point.All this said, last summer my family and I went on vacation with said cousin. On this vacation, he kind of just hung out in his bedroom, being antisocial. My brother and I, being the nosey guys we are, try to find out why our cousin is being like that. We end up stalking his facebook and finding out he was gay, and assumed that that was the reason he was not hanging out with us on vacation.So the next night, my brother and I went into his room and said "hey Joe, you know we really don't care if you're gay, right? We like you for who you are. Can you please hang out with us?"He ended up hanging out with us a little more than he was, but still mostly kept to himself.Were my brother and I correct in approaching him in the way? We didn't mean any harm by it.What would be the best way to approach a situation like this?I appreciate any feedback. Thank you.