Fear of rejection at your age? You have got to be kidding! What are you actually afraid of? That she will be rude to you? Don't you think that if someone is so shallow as to be mean to you for no reason, then she is obviously shallow with everyone, not just you? Aren't you a good person? Don't you make a loyal friend? Aren't you generally honest and trustworthy? If so, then if anyone rejects you, isn't that HER loss? She missed out on meeting such a great person such as you. And if she really is an immature person who cannot appreciate your unique qualities, then she did you a favor by rejecting you, because now you don't have to put up with her. Count yourself lucky that jerks don't want to talk to you. It saves you the time and trouble of having to reject them later on. Don't be afraid of rude people. They cannot harm you.
Also, consider this: every girl that you meet is a human being with the same fears and insecurities as you. If she is rude to you, it is only because she hates herself. Next time that someone is rude to you, just say, "I'm sorry that your self-esteem is so low that bullying other people is the only way to feel good about yourself", and walk away. You are better than they are.
It might help you to get over your fear of women if you make it a point to practice small talk with other people wherever you go; for example, next time you are buying groceries or ordering a burger, ask the waitress or the cashier, if the place isn't too busy, how her day has been so far, and actually make a small conversation while you are there paying for your stuff. It will probably make her day that some customer acted friendly, and you will soon see that talking to a girl is no more difficult than talking to a guy. Or just chat with people while you are waiting for the bus. If someone seems uncomfortable, then leave them alone and talk with someone else. Practice makes perfect.
Try watching this video (first link). Notice in the video, before the guy sat down, he told the girls that he only had a few moments to speak to them as he had to meet someone else soon, which sets a "time constraint" (meaning, he is letting the girls know before he even sits down that he is not going to bug them forever -- people are more likely to talk to you if they know that you aren't going to just hang them around all day until it gets awkward between you). Speaking of awkward, it always helps if you do not overstay your welcome. It is better to leave too early, and leave them wanting more, than to leave too late after they are tired of you. Walking away and talking to someone else also establishes the fact that you have other options besides her, so you are obviously not desperate.