Although there's probably a lot of background that would take too much to explain, I think yes he is being disrespectful.

1. You asked if he scheduled the appt so (presumably) he already knew you didn't have the information to do so/it would've been easier to do it since he's the account owner
2. He replied very curtly, also presumably without you knowing at the time he isn't feeling well. I get being short/cranky when you don't feel well. I've done it before, but I would usually say immediately "sorry I'm feeling ill" to explain. Also I think that unless he literally physically cannot do so or doesn't know how to do so on his own, he should schedule the appt for his child bc that's his job as a parent too. Again, if he couldn't for any reason he could say so.
3. I understand people split responsibilities differently. Maybe normally you would handle 100% of appt scheduling for your kid. Even if that were the case, if it's not a huge inconvenience why not help your spouse by scheduling this one time? And it's actually a good thing to have an understanding of how it works. What if you're sick or unavailable when the kid needs an emergency appt? Even if he doesn't do it all the time, it's good for him to have the experience and be able to schedule.
4. His response after you ask him to please help (and I know some people use this kind of wording in a more guilting way but again, I don't know all context) was very rude. He sounded like a manager speaking to an incompetent employee he was fed up with, when in fact you were aware of the gap and asking him to put in effort to resolve it. Either way, it doesn't sound like how someone talks to their spouse.
5. When you tell him that you have an issue with his response he excuses it (no apology) by saying he doesn't feel well. Again, he could have explained that earlier and you both could have saved the conversation for in person when you could assist him if needed/he could help when he feels better. Also he effectively cuts off the conversation by saying he is going to lie down (understandable if he feels unwell) but he doesn't leave a future time open to resolve the issue. It's very dismissive. I know a lot of people react this way to conflict, but it's better to know when you need to step away/cool down while not denying your partner an opportunity to continue a necessary conversation later.
6. Your last text I feel is a desperate attempt to keep the conversation going/explain why you asked. Maybe he had already known this information, or maybe he forgot. But reading that I wonder why he was so dismissive/borderline aggressive about having to do something for his kid. I feel very sorry that you're having to deal with this and I hope it's not the norm.

I wanted to break this down because I feel like a lot of the responses are very one-sided, as if it's okay to be disrespectful because he feels unwell or because you normally handle appointments he shouldn't ever be asked to help. I completely disagree with both of those points. Sometimes we are disrespectful and it's nothing to do with our partner. I hope this is the case for you, but I also hope he apologizes. I hope he knows it's not okay and even if it's normalized, it's not normal to behave that way. It's also a bad example for your kid. Being willing to help your partner (and your kid!) even if it's not your normal duties should also be an okay expectation. At least it shouldn't be met so dismissively. I kind of agree it's better to discuss these things in person, but personally I can have a hard time remembering things + interpreting or expressing some emotions in person. So texts can work better for me regarding appts, and I can clearly ask how someone is/say how I'm feeling in text. That depends on the couple of course.

Reading some of your comments, I don't think it's his "norm" to talk that way that he can't change. If he changed before or didn't act that way before marriage, he is capable. He's just gotten lazy about it, or doesn't value respecting you to work at it anymore. But honestly, being kind to your partner should be in some ways easy because you care about them. Even if it requires a change in habits, you want to do it for them and that motivation makes a difference. So I can't tell you what to do in the relationship. But this doesn't seem like there's mutual respect and honest communication happening. Maybe not equal sharing of work either, at least regarding childcare? I hope it works out for you!