Well, I suppose something will have to give sooner or later. Either you'll find eye contact gets less painful or you'll continue to find it so bad for you that you'll finally have to stop doing his bidding and he'll just have to accept defeat on that one. Can't you just stop going there and find another therapist? He seems awfully pushy and more certain of the rightness of his treatments than I'd think anybody ought to be. Or maybe you could write him a letter or something, if that's any easier for you than telling him in person to stop pushing the eye contact thing. Or maybe somebody else could have a word with him on your behalf.
I really don't think eye contact is so vitally important - it's no doubt useful if you can do it fairly painlessly, but I've got away with very little eye contact - I very often forget it, or find it distracts me too much from focussing on what's being said to me, and when I do remember it, I usually just shoot the occasional glance and look away for the rest of the time, which seems to do the job OK. In a lot of social encounters the detail of what's being said isn't that important, so that's when I risk it the most. But mainly I just don't look right into people's eyes, and yet I've managed to hang on socially. I don't know where this society is that expects eye contact so strictly. Most people I associate with are fairly tolerant about that kind of thing.