The important question is not which eye you can close, or which hand is dominant, but which of your eyes is dominant. For most people, their dominant eye is the same as their dominant hand and life is easy. For a few, their dominant eye is on the other side, a condition known as cross-dominance, and that makes life more difficult.
There are a few different ways to determine eye dominance, but I find this the easiest: hold the back of your hands out at arms length and make a smallish "hole" with your thumbs and index fingers. With both eyes open, put a distant object in the hole, i.e., so you can see it when you look through the hole. Then, keeping the object visible through the hole, bring your hands to your face while keeping both eyes open. You will bring the hole to your dominant eye.
The new conventional wisdom is that you should shoot from the side of your dominant eye, even if it feels a bit difficult at first. Thankfully, I don't have the problem so I can't speak from personal experience.
NRA Benefactor Life Member
Pemigewasset Valley Fish & Game Club