Of course it's not unreasonable to ask for references, so I assume that this is the real question you're asking:
Was I being unreasonable or abusive in any way?
I think that's a great question to ask, and well done for asking it. I hope that what follows doesn't come across as too harsh and that you can learn from it.
I've read the chat transcript and it's pretty clear that the level of aggression and confrontation rose pretty quickly. In my opinion, though, it was mainly you who were raising the level of vitriol, and that the OP was only responding to it.
You start off with a perfectly reasonable response for a reference. The OP responds with a reasonable sounding argument as to why they think that's unnecessary. Then we get the first of the more confrontational posts.
RA Well that renders your question unanswerable. If you cannot come up with a specific instance of what you mean by the very vague statement that "a Lie group has some relation to a differentiable manifold", then how can we help you? Why are you not prepared to cite a specific instance of an article that is giving you problems?
Instead of raising concerns about the question, you're now raising concerns about the OP themselves ('you cannot come up with a reference', 'Why are you not prepared to cite...' etc.) I'm sure you didn't intend to come across as confrontational, but I can understand the OP feeling a little attacked, which is why their first instinct was to double down on their position:
OP If you're an expert then it's definitely not unanswerable. An expert has the knowledge to understand the material in different contexts and thus especially has no difficulty with a mere layman question. If I cite a specific article then you will inevitably reduce the entirety of the question as being specific to that article and thus you will inevitably fail to address the generalization the question is actually asking about which is not and cannot be specific to that one article. I asked a general question, I'm fine with a general answer, that's exactly what anyone would expect.
OK, now they're making it personal as well ('if you're an expert'). But your response takes it to a new level.
RA That earns your question a close vote. I suggest you try to think about what math.stackexchange.com is for. If you think it is a service where highly qualified experts provide free tailored advice disregarding your arrogance or unwillingness to help them help you and correctly second guessing your ill-explained questions, then you have come to the wrong shop. It is a service where, perple with an interest in mathematics at all levels of experience and knowledge try to help each other. It doesn't sound like the right community for you.
Well that escalated quickly. This is quite a serious accusation: that they treat the members on the site like their servants who will just give them answer to their questions. I'm really not sure that that kind of allegation is justified here. You've also dismissed their question as 'ill-explained', called them 'arrogant' and 'unwilling to help' and even claimed that the community is not for them. I don't think you're in any position to complain about them calling you 'arrogant' or 'elitist':
OP You're being arrogant right now by being elitist. You want every question to look technical because that's what appeases your personal pride, you don't actually care if anyone learns anything, you're just an elitist. The reality is no one cares about that, this isn't some ground breaking paper we're writing here, it's a basic question that many people have. If you actually cared about providing material for the benefit of the public, this wouldn't be an issue for you. Any actually accredited "expert" is capable of addressing such a layman question as this.
By this point, the conversation has already become uncivil and vitriolic.
Then we come to this very question, which also suggests that you are unwilling to consider your own part in the dispute.
Is asking for a specific reference to an article mentioned in an MSE question unacceptable?
This is a straw-man fallacy: what caused the OP to react the way they did was not the (totally reasonable) act of asking for a reference, but the confrontational way you approached the subsequent discussion.
asking for the OP to provide a reference and got what feels like a a very vitriolic exchange accusing me of being an elitist in reply
This account of the conversation skips some very important steps in between which you should take responsibility for.
I'll try to walk away rather than rise to provocation next time.
In my opinion, the one who rose to provocation was the OP. They probably should have walked away as well.
I don't mean to make you feel bad about any of this, and I'm sure that none of it came from a place of actively trying to attack the OP. But I hope also that you can see why that might have been how it came across to them.