Tbh talking on this forum I get the impression that American culture puts a lot of weight on eye contact. I have never experienced a similar narrative about it. All I heard was more like don't look on the shoes, it makes you look shy or something similar. Firstly, no very negative connotations of poor eye contact, more that it makes you seem shy, secondly, nobody seems to bat an eye if it's not direct eye contact, but somewhere on the face or if you look in the direction of the person you're talking to. The whole narrative seems like an American cultural norm to me along with having to smile all the time, from an European perspective, Americans smile much more than Europeans do and also speak louder. Making very strong eye contact seems to fit the general norm of projecting a strong presence, as opposed to more toned down European norms. I can't speak about all Asian countries, but I'm interested in Japanese culture and they're even more toned down, more gentle in their behaviour. Apparently they avoid eye contact altogether. To wrap it up: it strikes me as a cultural norm how much eye contact is made. Even on this forum, I've never heard that sort of narratives around it before joining here.I think different self-aware creatures, such as anthropoids, will adapt to different ways to do similar things. Even among humans some cultures use high levels of eye contact, others much less so, and will be using other body language cues to express the same sorts of information from one to another (including emotional and non-verbal information).
To exclude DNA in looking at commonalities and differences is difficult, because it's the DNA that provides the ability to perform these complex interactions. It seems the better anthropologists and biologists get at studying simian social structures and behaviours, they uncover more aspects that relate very closely to ourselves, even though carried out differently (using the body language etc that works for that particular species.
I'd be inclined to suspect this is pretty mutable too, and has a lot of environmental factors. In some environments having high level eye contact is likely an advantage over other methods, and our brains are trained through environmental state. Someone from another low-eye contact culture has the same brain as their opposite number, but have simply developed in an environment that present a particular way of communicating as most advantageous in that environment.