I had the opposite problem. I made too much eye contact. I have PTSD, so I'm hyper-vigilant, meaning I am way too aware of my surroundings, and always on the look out for danger. I also have a hyper startle response.
I didn't even realize I was staring at people. It took me a long time to realize that was why people were uncomfortable around me. Once I did realize it, it felt strange, and was always worst walking past people in the hallways at work. I didn't know if I should make eye contact and say hello, or look at the floor, or avert my eyes somewhere.
When I did first start averting my eyes, it was so awkward because I didn't know where to look, and people sensed my awkwardness and felt awkward around me. I became extremely aware of how much eye contact I'd make. For me, I have to err on the side of too little eye contact. I'd rather appear shy than freak people out by staring at them.
It's getting easier now. It's been 3 jobs later, and I feel like I'm finally not freaking people out. I'm learning how to look away, avert my eyes, and look natural about it.