At my new office, I got to know a few people and among them is a woman ("Alice" for simplicity). I could call all of them acquaintances including Alice.
I usually greet Alice outside of work, with a smile or "How are you?". I usually do not greet her while I am at work. I say this because her desk is just after mine, she passes me each time she comes back in after taking a break and it kind of feels weird to smile at her each time.
Today, it so happened that I finished my work and I was looking for my supervisor, so I could inform him that I was leaving for the day. He was in a meeting, so I kept looking behind me to see if he was done. While doing this, Alice left her desk and walked in front of me to . I did not greet her, even though I caught her eye, I just turned back to my computer screen awkwardly and I kind of felt like she was expecting me to smile or acknowledge her because she turned away awkwardly too.
I felt terrible after this. Most of all, I felt guilty that I didn't smile because I thought she expected it.
Should I apologize to Alice for this incident?
Or am I reading too much into this and let it go?
What is the best way to handle such a situation?
Upon @Catija's suggestion , I would like to add that I usually don't greet Alice during work because it might be taken as a sign of romantic interest by other colleagues .