Making eye contact with a Grey Heron and just a matter of weeks later a Tawny Owl were profoundly moving experiences. Getting close enough to these wild birds to see actual eye movement without them immediately fleeing is a privilege in its self but it taught me a lot about my sense of self how it relates to the natural world. Very humbling but these days i walk about with a husky who wild animals are rightfully scared of though we did sneak up on a baby rabbit basking in the Good Friday sunshine (i honestly thought it was dead) that did not run off till i was literally looking down on it. And today she was in a small brook drinking when a kingfisher flew past, just inches away from her head and feet away from me, the brightest, most colourful, striking thing you could ever wish to see in nature and it's almost right on my doorstep.
As for as domestic animals are concerned i don't know how you can't make eye contact with them. When i bought my dog i had originally paid a deposit for a different dog but one of the pups had heterochromia and when i went to pick her up i changed my mind and took home the one with one blue eye and one brown eye. She often just sits and stares at me occasionally winking, i often wonder what's going through her head when she does this, though she is probably thinking, what the hell are you looking at?