Detecting UV by humans
First to address some of the comments:

Concentrated UV damage does hurt your skin, it's called sunburn.  Lighter amounts of UV damage can cause cancer over time.  UV damage can be "felt", as can photonic pressure in extreme cases.  The issue at hand is "what is detection."

Back to your eyes:
As addressed in this answer, the human retina is able to detect UV light if it is not blocked by the lens.

I believe your question, restated slightly, is "can we call the pain experienced by UV damage to our eyes human detection of UV light."  Generically I think the answer is no, because we lack the specificity to determine that it is UV light. Many forms of ionizing radiation can lead to pain, and we would be unable to distinguish between them.  We cannot tell the gradation from UV to gamma, despite the damage they might cause.  Contrast this to the visible light spectrum, where we are quite good at determining the wave length.

Further, you could have similar damage caused by infection, where again you couldn't distinguish between the pain and fuzzy sensation caused by the very beginning of infection vs slight UV damage (though blueberry might help).

Does UV light damage your eyes? Absolutely in large enough doses.  Can it be called detection?  Probably not.  We can deduce UV damage from lots of things, including changes in the color of the pigmentation of your eyes (gold flakes = bad).  When we get to the level of deduction, however, we might as well be using machines to detect UV light and save our eyes (and skin) the trouble.