A bit of context, this girl and I act like couples but it's not 100%, even though I told her many times I liked her she probably doesn't take it too seriously. She literally acts like we're a couple by holding my hand, going out on dates, and even once gave me a kiss but despite all of that she out of nowhere said "ew, we look like lesbians" and whenever she gets asks if she's lesbian she says no. After all of that, I got tired of her behavior and asked her if she could show some respect towards me because it seems like she thinks of me as her little puppy. She then started to ignore me and so did I. Us ignoring each other lasted about a week so she came up to me and just said "hi" and try to act as if nothing happened, so I was waiting for her to apologize or say something about the matter but she didn't and she then left. Now, we're still not talking but I can see her staring at me from afar. What should I do? I was thinking that I wanted a bit of closure by talking to her properly but I also want her to be the first one to come up and actually talk about this because it's always me chasing and I'm tired of it and I just wanna see if she actually cares.
This is the sort of question that without your age, background and a timeline on your question, no one can answer. In very general terms, all relationships start as friendships and discovering what you have in common. Importantly, this includes your tastes in everything from food and music, to whether you prefer an evening at a nightclub to a country walk. With sufficient common factors the friendship will continue, without them you will drift apart. If things move on, you need to take them slowly. For a friendship to develop into a REAL relationship it takes time and their are a lot of hurdles to cross. For what it's worth, my suggestion is to take less notice of the day to day and see what transpires over the next few months. Do not build up false hope, but at the same time do not burn bridges that don't need to be burnt.
From a bi man, accept that women are much more difficult to 'understand' or whatever than men. Guys want sex, women want...?
Walk right up to her, tell her you are very interested in her ----that you are a lesbian and that if she feels the same interest----then tell her "lets date". Sort it out and date or move on with no hard feelings.