This is just rednick speculation, but here goes.Deer can obviously see well enough to determine shapes. For instance, a human-like shape at a distance will cause them more alarm than will something that is shaped like a plant of some sort (or even another deer). If they can do that, then it isn't silly at all to think that at close range the shape of a human face could be alerting to them when that face is a shade that is different than your surroundings. Deer then watch these more troubling shapes and if they don't move then they don't necessarily alarm, but still they are treating these certain shapes differently. I've heard it claimed that the head and shoulder shape of humans is different than other things in the woods and that ghillie suits breaking this up is a big advantage of them (but who knows).
A study to determine if deer recognize human shapes and are bothered by it would have to be properly designed to find the effect. All studies start with the idea that the effect is absent and then you collect evidence to disprove that null hypothesis which then infers the effect.
Camo that breaks up the human outline is praised. This might be wrong, but it infers that deer see shapes and are troubled by some of them. Seeing just the eyes (or just the whites of them) might be a stretch but I would not find it odd if the whole package of the human face (including 2 forward facing eyes in the right location) might be troubling. Heck, there are lizards with spots that look like their eyes on their tail and this makes predator attack the tail.
I would say a deer probably has a hard time detecting a face. I say this merely because I have 4 deer on my wall, and all of them have a vivid white throat patch, white around the mouth, white around the eyes, and white around the ears. They also have their namesake white tail. Why?
My best guess is to help pick up on nonverbal cues. A direct stare. Raised and alert ears. Laid back and aggressive ears. That kinda thing. We have eye whites to help us pick up on those subtle eye movements that convey whole paragraphs in a nanosecond. We couldn't pick up on that with solid brown eyes like a deer has, so nature helps us out. My two-beer theory is deer similarly couldn't pick up on those nuances without a visual aid.
Compared to a deer, humans have very flat, nondescript faces. Just a teeny little nose and some beady little eyes recessed into our head so the sun doesn't burn them out. The whole face is one color. If a deer can't perceive another deer's long and exagerated face without an aid, I don't think they can pick up on our little pug faces. I do believe if you're a Caucasoid, you're disadvantaged because you're basically alarm-colored to them. White is the color of a dominant deer staring directly at you or a bunch of your buddies running away. A moving flash of light is probably similar to a blinking red light to us. We notice that because it's never good. But as far as having the visual acuity to distinguish a face from say, your other cheeks...I question it.
Other animals do have eye spots. For a butterfly or lizard to have eyespots makes sense. Birds can most definitely see eyes, and staring eyes are not food. Fish have tail spots, which can be handy because most pescatarians prefer to eat fish head first. Strike at the tail and you're likely to fall short on your strike, or now you have to swallow the fish tail-first and choke on spines, allowing for escape.
I think it's natural to assume deer see faces because we obviously see faces readily. We see them so well we see them on the moon, in wall plaster, on our wall sockets, and on pancakes. But we see them because they're absolutely vital to our well-being. If you have to live your life taking other people's words at face value (hee hee) without being able to cue in on body language, tone, and facial expressions, you're gonna end up disadvantaged because people lie like crazy. We lie so much that we make whole games out of it. Acting, stand up comedy, poker, etc. Games are a safe way to practice a species' serious skills. Human faces matter to humans. Deer? I don't think they're as essential to perceive. Movement gives them the info they need, and it's less costly to evolve the ability to perceive that.
It all rolls back to our habit of anthropomorphizing deer! We know that we freak out when strangers stare at us or ours, and we assume they react to that stare the same way we would.