From your description, I can see that you want to keep your eyes wide open while swimming in a swimming pool right? So, as a matter of fact, if you swim in your personal pool, it is ok. However, you should know that in some public pools, there are some chemical and toxic stuff in the water, which could lead to some eye infections and uncomfortable feelings. Therefore you had better wear swimming goggles.
I think it bad to keep your eyes open when you put your eyes underwater because people usually add bleaching powder to the water in order to sterilize, and the water in the swimming pool can contain blue vitriol with the purpose of preventing aquatic plants. The bleaching powder, the blue vitriol or other additives in the water can stimulate eyes. Moreover, the surfaces of the eyeball can secret fluids to lubricate and sterilize, but it merely functions in the air and the protective mechanism does not work underwater, under such situation, all kinds of germs will intrude into the eye. So I advise you not to open your eyes or to wear swimming glasses when you are underwater.
Normally people could open eyes under water, but whether it is harmful is depending on what kind of water you mean by. If it is absolutely clean, it should be ok to open eyes in it. There is such method and production medically to help people wash eyes. However, if you are trying to open your eyes in swimming pool or someplace like this, I am afraid it is not a good idea. Water in swimming pool contains various bacteria and disinfectant, which could be very irritated to eyes. Although ears produced by eyes could kill 90-95% bacteria in eyes, yet there is still 5-10% left which can't be killed. Many people got conjunctivitis because swimming in public pool. I, myself, could help closing my eyes in swimming pool when I was under water. This is the so called unconsciously active response to protect eyes. wearing swimming goggles is much better than not.