OP, it sounds as though you don't expect your husband to be able to do things so you ask him to do things expecting him to fail and then get annoyed when he inevitably proves you right.
Why didn't you either call and make a telephone appointment for the doctor to call you, or ask your husband to call and make a telephone appointment for the doctor to call him and have him sort the whole eczema thing out?
Asking your husband to call and make an appointment on your behalf is only going to save you a couple of minutes even if he does the job perfectly, because you still have to ask him to do it, and then the doctor is still going to call you and speak to you about the eczema problem, and then doubtless you are the one who is going to be following up on the doctor's advice.
If you ask your husband to do 10% of the job and leave you to do the other 90%, then if he fucks up that 10% the whole thing is going to take you longer to sort out than if you'd just done everything yourself.
If you asked your husband to sort out the eczema problem, then he would have had to call and book the appointment, and then the doctor would have called him back to discuss it, and it would have been his problem if he hadn't checked that they'd got both children's details and the doctor was only willing to discuss one child, and his job to book a second appointment to discuss the second child, and then follow through on the doctor's advice.
At some point you're going to have to decide whether you prefer to do everything yourself to ensure it's done properly, or let go a little bit and let your husband do his fair share. And I mean actually do his fair share, not just do little things that you delegate to him occasionally.
Because the way you're asking him to do things like this, it sounds like you're making a cake and asking your child to come and put the raisins in. You're not asking him to do something to actually take the burden off you, but to give him a job to do. And that's pretty pointless unless you're willing to give him a whole job to do and let him deal with the consequences if he doesn't get it right.