Ask Mr. Smarty Plants

Ouch! We are so sorry. We are gardeners, not doctors, so it sounds like you did the right thing initially. The Canary Palm is native to the Canary Islands (thus the name) so Mr. Smarty Plants does not know much about the plant itself, since it is out of our realm of expertise, that being plants native to North America.

But we  did search around and try to find the answer to your specific question and found no reference to allergies or poisonous parts. Here is an article from Dryad Musings on specific damage that can accrue from the plant, apparently from personal experience on the part of the writer. From Dave's Garden, here are some pictures and descriptions of Dangerous Palms, which scared the daylights out of us. Scroll down the page to find pictures and discussion of Phoenix canariensis.

We feel (totally a non-educated guess) that your eye was really the target of blunt (or rather, sharp) trauma from the thorn and, personanally, would advise you stay with medical advice from those that are professionals. If you didn't get the kind of results you hoped for from your first doctor, we would suggest you seek a second opinion, but not one from plant people. Our only valid advice would be to stay away from the darn things.