Just throwing this out there --
Do we really need to think of eyes closed/open in such a black and white way?
In my shooting (bow, shotgun, or rifle), there are relaxed times when both eyes are open and I'm seeing clear and feeling good. Then, there are times when I've got to squint a little or really bear down and close my non-dominant eye to clear up a sight picture. As a shooter, I think it pays to be able to shoot either way in most disciplines.
In general, for shotgun I'm both eyes open but will sometimes squint the non-dom eye. You get more field of view this way, which is cool in wingshooting since if you're doing it right you are focused on the target and just vaguely aware of the barrel's relationship to it.
For bow, I tend to really squint or close the non-dom eye entirely. Better precision. Like 5MilesBack, I tend to see a double image with both eyes open. Not good for bow!
For rifle scopes and red dots, either way. Sometimes both open and let the dom eye clarify the image. If that's not working in the moment, I'll squint or close the non-dom eye as a kind of focusing tool.
For what it's worth, I'm left eye dominant, right handed. Always been that way even as a kid. Used my right hand for tactile things like writing, throwing a football, eating spaghetti, etc. But when hitting a baseball or shooting, I always did those tasks left-handed since the eye matters a lot in those disciplines.
Now, I shoot rifles, shotguns, and bows (mostly compound but experience with recurves) and even used to coach these things through work.
End of day, I'd say the proof is in the pudding. Try various techniques out and methodically track your scores and hits -- when are you better?