Yes, sunlight can hurt your eyes because the sun contains UV rays and if it is absorbed directly by the eyes, it is very dangerous. This can lead to many UV-related illnesses, such as cataracts, skin cancer on the eyelids and macular degeneration. What's more, if you have blue and green eyes, you will be very sensitive to the sun because of less pigmentation. Pay attention to that and avoid watch the sun directly to for a long time.
It is not strange at all but very normal and reasonable. You know, the strong sunlight can burn our skin, while you don't know that it can do the same to our eyes. There are UV rays in sunlights which is very hazardous. What it can do is more than just darkending our skin. It can burn the skin, cause wrinkels and spots, and even skin cancer. As to impact on eyes, if long time exposure to the sunlight, the surface tissue of eyes would be burned. As to more serious problems, it can cause pterygium at the corner of eyes, pinguecula and cataracts. Long term exposure to the UV radiation would increase the risk of developing cataract or accelerate the growth of cataract. All in all, it is much better to wear a good pair of sunglasses which can provide 100% protection from the damages of sunlight. There is no reason for you to refuse the sunglasses.
If you find that your eyes hurt when you look at sun, you should observe whether there is the problem for your eyes. Actually in the common way people will not feel any hurt when they look at the sun without strong sun light. Your eyes may get infection because of the coming bacterium. You need to use some eye drops to release the symptom. If there is intensive sunshine, you may feel difficult to open your eyes. If so, you need a pair of sunglasses.