Actually even if you close your eyes under the strong sun lights, the uv rays will come into your eyes and damage your eyes vision. You should avoid doing so. If you really want to have the sunshine, you could wear the sunglasses which will help you protect from the uv rays and other bad lights. You could take this into consideration and buy the sunglasses with cool look.
Well, as you said, staring at the sun is one of the most difficult things for any one of us, thus, we should avoid doing such stupid things. Of course you could face the sun with your eyes closed but you should know that would not make the situation much better, you should get a pair of strong sunglasses to do this. Anyway, who likes that thing? Just take care of your eye health.
Yes, generally speaking, the intensive sun can definitely hurt our eyes by its UV light or other invisibly dangerous infra-red or ultra-violet radiation. We all play such trick as bring the sunshine to a focus by a magnifier to burn a paper or something like in our childhood. It is also true to our eye imaging. The light can form a focus in our retina so that we can get the information of objects, then transform that message into images. Well, when the light is too strong, retina will be burned. It may not present at once, instead the symptoms occur in few hours later. When you close your eyes, your pupil will keep opening. Once your open your eyes after facing the sun, your eyes keep opening and the light will flood into your eyes sensitive part more than before because your eyes can not react to the strange scene quickly and shrink their pupils. Then, you will have symptoms like seeing many small black spot in front of you, as well as dizziness. All in all, donâ€™t do it, if you actually want, use goggles.