The phrase “my eye” isn’t used much in english speaking countries anymore, but you do come across it in older books and movies and I’ve always liked it as a way to show sarcasm/disbelief (e.g. “Him a gentleman? My eye!”) What I didn’t know was that this shows up in both French and Japanese as gestures with a similar use.
In Japan it’s called Akanbe and pulling down the lower eyelid to show the red is usually combined with sticking out your tongue. In France you might say “my eye” – mon œil – while pulling down the lower lid. Both are sarcastic and, I think, more a thing younger people do.
I’d always assumed “my eye” was like “I’ll eat my hat” i.e. something you’d sacrifice if proven wrong and I thought it must relate to something like Odin’s story in Norse mythology, but maybe it’s just that eyes look kind of gross when you see the pink bits?