No.
I can't. My non-domiinant eye is also my poorer sighting eye as well. Although I don't wear glasses, my left eye is just not as good as my right, which is great.
So, although I shoot with my left hand, and use just that hand with it out-stretched, (I shoot one handed), I turn my head to the left to get my right eye so I can use it to sight down the barrel.
I, too, keep both eyes open. It is easier and less effort on my facial muscles, believe it or not. My face gets tired scrunching up and trying to keep closed an eye.
Because of my right eye being the best, I am learning to shoot with my right hand when shooting the rifle. That by far is the weirdest and most foreign to me, grabbing the gun with my right. It seems like someone else is holding it. That is what is taking the most getting used to.
Sitting down and holding the rifle with my "wrong" hand is slowly getting okay. But to grab it and raise it if I ever try to shoot standing up is still something that I have not gotten used to.
cavman