It doesn’t seem so much that you ‘need to convince someone that gay marriage is wrong’ somehow, as it does that you
want to reassure yourself that you’re right. If you were convinced, wouldn’t you have already educated yourself on the usual excuses as to why it’s an abomination? The sterility/menopause objection is such a basic one that you’d have had it covered as best possible if you’d had any interest in the subject before.
It’s not about your ‘friend’, if one can call him that – at least you sure aren’t being much of a friend to him. It’s about
you. You’re uncomfortable with the existence of opposing points of view, but uncertain as to how to deal with them. And so, not having anything else to go on, you try to steamroll it – and you may take my word for it that you’ll fail utterly if you keep going this way.
You think he envies the rights and privileges afforded straight people? Perhaps he does (hell, I would, if I’d fallen in love with someone with a matching set of bits), but you – you’re hardly an innocent yourself; the Church is a hospital for sinners, is it not? Your own sins and shortcomings seem to color your perception of him: pride, in that you will not accept a debate on equal terms; cowardice, in that you run for help to people whom you think may know better how to argue than yourself, instead of facing the question with what courage and conviction you might have; and willful ignorance, in that you refuse to consider your acquaintance’s motives and beliefs at face value, instead imposing your own ideas of what his ‘agenda’ is.
Give up. You are not going to win anything, and you’re only demeaning yourself by trying. The best result I can see from this is that you will become estranged from someone you call a friend. Maybe later you might find it appropriate to talk about the subject – but not until you have done some research and soul-searching of your own.
Know why your faith frowns upon other forms of marriage instead of begging for help when you get put on the spot. Understand the difference your faith places between a sterile heterosexual couple and a homosexual couple – even why, to Catholicism, that’s a red herring from the start (between you and me, they’re wrong to consider it so, but… try to figure out why that is for yourself

). And above all, watch out for that plank in your eye.
St Francis of Assisi said ‘preach the gospel always; use words when necessary’. It’s a lesson a lot of Christians could use. Were I you, I’d take it to heart.