I’m sorry you’re going through this. There have been lots of responses and I haven’t read them all, but there could be a few things going on with your coworker.
The first thing that comes to mind is that she treats you this way and not others because she is getting a reaction out of you. Especially if it’s subconscious to you, if you show averted eye contact, a slight sigh when she doesn’t return your “good morning,” a wince, your body braces up when you see her, etc. she can be taking that to fuel her need to bully. People can pick up on that, some easier than others. Because of your sensitive nature, some people may mistake that for weakness and there are people who prey on the weak. Always stand tall, keep your head up, and carry yourself like the dignified daughter of Christ you are.
She could just be dealing with problems or insecurities of her own (see above how some insecure people deal with that). Is she really sighing because she’s reading the email or just sent or was it coincidental? Maybe something happened to her, or her best friend used to work with her/sit in your spot, etc. and she resents that, not you. You don’t have to speak to her every day. That’s not rude. She may need space.
One thing, though, someone not speaking to you doesn’t necessarily warrant a HR visit. She hasn’t done anything substantial to warrant something like that in a corporate world. If she is bullying, however, that’s a completely different story, but you would still need sufficient proof (rudeness can be, simply not speaking to you can’t).
She may just feel your personalities don’t mesh. That’s fine, not everyone’s does. You don’t have to prove her wrong. It may be a good idea to try to ask her to lunch or offer to go with her to a noon Mass sometime. She may find that you are a good and sweet person. She may refuse to go or not ever see you that way.
Maybe you’re confusing your own insecurity with sensitivity, you just don’t act rudely like she does. Instead, you turn inward, wondering what you’ve done wrong. Nothing. You haven’t done anything. Just remember that she does not, nor does anyone else, own your happiness. Don’t hand that power over to her.