The argument is that eye contact is a sign of aggression or dominance in many species. This would indicate that these animals can perceive eye contact. This is true in a sense - in that the animal already recognizes the animal staring them in the eyes as said animal. The problem with this is two-fold, and a matter of semantics.
First, if the deer recognizes you as a person, with eyes, and are making eye contact, it's too late for it to matter for hunting purposes. At least the deer I hunt - once they realize you're a person, an arrow better be on it's way. Otherwise they're on the move.
The second issue I have with it is that we can perceive eye contact at close distances, and do it very well. You can actually assess dominance hierarchy in humans in a matter of seconds with eye contact alone. Measuring the time that each individual makes eye contact with the others when speaking and listening paints a really clear picture. Unfortunately, put a distance of 10 or 15 feet between you and I, and you can't tell if I'm looking directly into the your eyes, or at your nose, or ears, or mouth, or neck even. What you can perceive is that I'm looking at YOU. That's the semantics. Eye contact means I'm looking at you - Sure, it is more poignant at close distances when you're staring directly into someone's eyes. But being looked at generally is what eye contact is. There isn't a "sense" that can detect eye to eye contact. There's just a more precise image depicted in the brain at close distances. You sense the person in the crowd making eye contact because they are looking at YOU, not necessarily right in your eyes.
A last detail I'd toss in is that its likely that deer can't even see your eyes when you're 20' up in a tree, and backlit. Your face is going to be dark, and your eyes darker. From what we can tell so far, their vision wouldn't allow them to pick up the direction the eyes are looking. Again - they have a prey instinct of noticing a face that has binocular vision staring at them. Back to my first point - if you've been identified as such a critter, it's too late.
So I'll take a slight middle ground here and say that looking away might help them not pick up your facial features and identify you as a predator.
However, I'm willing to bet if you could somehow mount a set of mechanical eyeballs on the side of a tree, with no human scent, that could track a deer's eyes when it was in front of the tree, the deer wouldn't give two flips.