I’d put in a vote for not going so as to not help your uncle make part of your cousin’s deal be all about you – because it should be all about your cousin, if it’s about anybody, or better yet all about the solemnity and/or joy of the occasion. And sending a note or flowers saying best wishes and congrats and you’re sorry you couldn’t be there, but not necessarily explaining why not.
I agree on all 3 points.
How about:
“I like to be with Mrs. Rhymer when she’s in the hospital, especially when the staff’s minds might be elsewhere-- like the Superbowl.”
If your cousin was doing standup comedy as the marquee performer for the first time, but you were worried that the opening act would call you out and make fun of you, would you go?
Is this the cousin who had his daughter in her room without supper because she decided not to go through with her confirmation or something of that nature? If that’s the one, I certainly would NOT go. If that’s he and his family treat people who don’t share their beliefs, then why would you want to be a part of that?
(And was he the one who claimed God healed his diabetes, even though it was getting worse? Or was that some other Dope altogether?)
I think Tabula Rosa’s excuse is a perfect one!
Since you and Uncle are are speaking terms, call him up and tell him that you would like to attend in support of your cousin, but don’t want to do so if he is going to single you out on this occasion. Can he let it go just this once? Either that or sneak in towards the back. Old uncles can’t see all the way to the back row.
How about going and wearing a dress?
If you were very close to your cousin growing up, you should go to support him. Otherwise you should not feel obliged.
My parents are missionaries, i grew up in the church, am a leave me rhe fuck alone agnostic, attending my father’s retirement service is the single time in 25+ years i’ve even been inside a church – and yet i would go…
you’re not a jerk if you don’t attend but on your deathbed i bet you’d rather appreciate the all too rare family time than regret attending your cousin’s first time as a minister in his church
Would you go if the two issues you mention (alter call, hateful church) were not a problem?
If you still wouldn’t go, then send Cousin a congratulatory card and move on.
If you say, “sure, in that case, I’d go,” then you should go. You’re an adult and it’d be for Cousin. You are not joining the church and you can ignore the alter call (if uncle says anything, respond with “It’s Cousin’s day. Let’s don’t make it about me.”) But if those two issues are the ONLY reasons you won’t attend, yeah, that’s kind of jerkish.
Very sorry about the above. I missed the part about being called out by name. You ought not to go, but otherwise be supportinve of the young person. He will wise up at some point.
Isn’t it against the law to treat your sister as a cousin? 
I don’t give much of a rat’s ass about my cousins (who are all Southern Baptist douchenozzles who cover their racism and homophobia beneath a thin veneer of piety (not that all SBs are like that, but these are)), so I choose to have little to do with them. But it sounds like your situation is different, and you have a different relation with your cousins. Nevertheless, I agree that this day is about your cousin, so if your disagreement were confined to Church policy, I would say go. But the fact that your uncle seems to like making this about you would push me over the line into saying “Fuck it” and not going. Your going to the investiture is a statement of your support for your cousin’s life and choices which they know you disagree with; if they cannot make the same statement to you, then that is disrespectful and a bit hypocritical of them IMO.
On my last Sunday back home over these past Christmas, I asked Mom whether she wanted to go to 10:30 Mass at our parish or 11:30 at the nuns. She said 12:30 in our parish. I said, “OK, then I’ll go now at 10:30, see you later.”
“But I want to go later!”
“That’s fine, Mom. I’m not saying you have to come now. But if I go to 12:30 I’ll come out once more wanting to wring Antonio’s neck, and I’d rather not need to have confession on grounds of ‘I want to murder one of your convent-mates slowly with an extremely rusty and tiny teaspoon.’”
“Oh. OK, yeah, well, I guess you do have a point… sigh You know, they’ve told him repeatedly not to do that, he just ignores them.”
“Yeah, well, only cos he’s a recalcitrant ass doesn’t mean I need to listen to him fart. See you in one hour!”
Going to Church isn’t about coming out feeling like the world is a happy fluffy cloud, but it shouldn’t be something that leaves you feeling murderous. So, ego te absolvo, Skald.
My cousin? Sure. I’d be free to kick his ass afterwards, after all. 
No, it’s a different cousin. I have a LOT of cousins. I’m sure the morons in question will be at the ceremony too, though, and that their daughter (my first cousin once removed) will get the same pressure to stand up and be saved, sanctified, and filled with the Holy Ghost as I’d get.
And the diabetic is my brother-in-law, who was refusing to get treated till his little girl turned on the waterworks. Little girl waterworks are very powerful to daddies.
Familial obligations are certainly important, and in this case, wife-in-the-hospital trumps cousin-being-invested.
I’m sure everyone will understand.
Don’t go. Tell everybody it’s so you can be with your wife in the hospital.
ETA: In my world, “immediate family member in the hospital” automatically trumps any and all other social obligations.
I don’t like using her as an excuse, and my sister knows I’ll be seeing her that morning, not that afternoon, as she’s coming with.
Don’t go. You don’t have a close family-type relationship with this cousin and the uncle is actively antagonistic toward you.
If your sister refuses to see that, that’s her problem.
This isn’t just about you. Here’s what you need to consider: how important is it to your cousin that you attend? Does his joy for having you present outweigh your potential discomfort? You’ve said that your uncles potential actions are forgivable, so that makes me think they’re not a show-stopper.
You also should consider the goodwill you’ll generate with your other relatives.
If you can’t attend, you should call as others have suggested.
~ ~ ~ ~ ~
Slightly off topic: altar calls seem to me to be totally intrusive and rude. In the churches I’ve attended, one’s salvation (or not) is a personal matter between the person and God. If someone were to try that on me, I’d say something like “the state of my soul is between me and God” and not elaborate further. And I’m a believer. If you’re not, a statement like that is non-operative and so not a lie. Think of something to say to diffuse the situation without upsetting their sensibilities.
As soon as the word ought is invoked you know you’re dealing with an argument that has no substance. Sorry Paul, I don’t buy this one at all.
Skald - if you don’t want to go then that’s all there is to it. It would probably be nice to send him a token of your well wishes to mark the event (followed up by debaucherous sex with your wife or not, as pbbth suggests) as it’s clearly important to him, but it sounds like you really don’t want to be at this thing, so I suggest you not go.