OMG
You are NOT making him ill. He is guilt tripping you because he can get away with it and you are a good person and want to believe in him and what he says.
But he is talking bollocks. Really, X used to do this too.
He is being a drama queen, a wuss and an unmentionably selfish tosser.
I remember when such responses would make me feel ill, too, sick to the pit of my stomach. It's because you make the approach as a normal, rational reasonable person expecting the same (I mean a printout FGS!) and they come back with this arsenal of grief which knocks you sideways.
When you think of all the responses you could have had:
"Sure, give me a minute"
"Arg, I cant right now but I'll do it asap ok?"
"Sorry, too busy, will have to wait"
What he did was completely OTT, and the worst thing is, he has trained you to respond that way, he knows exactly what he's doing. Can you imagine him behaving like that in front of an employer? Or his friends?
The only response he deserves is an eye-rolling "get over yourself" and a bottle of smelling salts (for when the poor fragile soul has another funny turn.
I know you cant do this right now, but that's what happens you start to detach and see them for the immature little weasels that they really are)
Look after yourself, and dont worry about him, because that stressful way you're feeling? That's because you feel wronged and KNOW you should be angry, but you're burying it as you can't express it right now.
Is he a narcissist by any chance?