As a decades long archery coach and avid shooting sports enthusiast (see my avatar and user-name), I can definitely state that eye dominance can be a very real hindrance for aligning sites for both firearms and arrows if one has cross dominance and hasn't addressed it. I've honestly never noticed any impact in a golf game, though. I am very right eye dominant which is great for my shooting sports participation (especially archery!) as I am also extremely right hand dominant. I could see eye dominance issues as a factor for aligning puts, especially using a putter shaft or fingers, or maybe even aligning shorter iron hits. If you putt right handed and align with your left eye, I could see it potentially impacting how you see the line differently while standing behind the ball or above it. From a long-hitting perspective, I would think that there's enough distance to not really throw off the perception. Plus, throughout one's life the body has become accustomed to make adjustments with basic eye/hand coordination, and I would see coordination being a much bigger issue as I've never had a problem with my extreme eye dominance.
I'd be curious to see what an ophthalmologist says, but I just don't see a big impact to the overall golf game.