I got introduced to this person in my neighborhood by a friend, and for a couple of times I said hi to him out of politeness. He also does not have a very close relationship with my friend. I don't have any interest to keep greeting him in passing so I just stopped doing that by avoiding him or pretending not to see him (I have little bit of social anxiety which could be a contributor). Similar situations has happened before. I would like to know if what I do is very inappropriate? And if so, are there any better options?
It would be odd if your decision is sudden, that is, if you had said hello yesterday and days before, and if you pretend that you don't know the person when you see them tomorrow, it would be odd. They might take it as a sign of something wrong they did to you, and start wondering why, or maybe even come up close to you to make sure if everything is alright.
It should die away gradually, for example, hellos should change into quick smile or nods first, and a few days later, just a glimpse at each other, and gradually you'll be back to being strangers.
I met a Watchman of the adjacent building to mine. He was nice and all, and I would often meet him on the way to my office and back. Initially I was okay with some quick hello and how are you talks, but it's not really to my liking to make it a routine.
Over a few days, I reduced it to just smile, or decided to walk through another route, which is fine by me.
And sometimes, I'm running late to work, so I don't even face his direction, but he'd understand I'm kind of busy.
I do not think "inappropriate" would be a term I would think of. What you have to consider though is what is considered normal etiquette where you live. It also likely matters if you are living in a large city versus a smaller community. It can change how that is interpreted considerably.
I live in the USA, Mid-West. I live in a small town. As such, here it is seen as odd behavior in an unfavorable light. It is seen so much this way I can easily give you an example. People that come to my home often have seen most of my neighbors outside enough to know who is and is not a neighbor. Last week I had a friend visiting. I went in the house to grab drinks and came back out. My friend asked me what is "up" with my "weird" neighbor. I ask what he means. He then goes on to tell me that this person walked past my home and appeared to purposely look at the ground as he walked by to "avoid" saying hello. I have no doubt he did. That is how that man is. It is seen as weird here. It causes people to wonder about him. They do infer something is "off" about him because his behavior is "off" to the common culture.
So walking by my house while purposely avoiding eye contact and avoiding saying hello isn't "wrong", and it's not even inappropriate. He is perfectly allowed to do so. He does so though while perhaps being unaware it causes people to have an odd view of him. This could be a problem for him in some cases. If something happened in the neighborhood, he could be someone they would automatically mistrust. When someone seems so purposefully closed off and like they actively seek to avoid contact, people are naturally prone to more suspicion of that person's intentions. And the thing is, this is probably the 5th or 6th person that has asked me that same question over the last several years. It's not one person's interpretation. And I have had other neighbors mention it as well, asking me what is "up" with him and have I ever spoken to him, etc. Honestly there is nothing weird about this man that I can see other than his obvious efforts to avoid speaking to anyone. I am not saying it's right that is how he gets viewed. I am merely sharing how culture matters in this regard and what other people expect socially.
Once somebody is introduced to you he/she becomes an acquaintance.
It is considered polite in many cultures to acknowledge an acquaintance when you meet, whether by nod/ smile/ wave/ brief greeting/ handshake/ brief chat, so you need to consider what is the typical behavior in your society, as many earlier answers have pointed out.
If you feel like ignoring the person it is your choice but they might misunderstand.
I have a bit of social anxiety myself and tend to avoid acquaintances, but I cannot ignore them when we come face-to-face. My typical solution (does not work every time) is
smile, wave or say hello, and keep movin'.
I think we all had some similar situation. What I do is always say Hi, in this case I know I'm not being rude or somehow offending him. If he wants he can start a small talk, otherwise he will just respond Hi and move on. At the end it's just a quick exchange of 2 words, nothing awkward, but it still depends on the level of your social anxiety. Anyway, I think that a simple Hi is never a bad thing