It sounds as if you are feeding the situation and mentally turning it into a problem. Sometimes parents can make a situation worse without even realizing what they are doing.
Your daughter is continuing to complain because she is getting a positive feed back from you, weather you are aware or not. Complaining for her is bringing her a very desirable result. Attention, More time with Mom, Love! What ever it may be for her, she likes it.
Try rewarding her after she eats a food she is afraid off. If she is afraid of ice cream consider that a good thing. Ice cream IS bad for people. If she just wants to eat with mom, let her eat with you. If she doesn't want to eat certain foods feed her some thing else or REWARD her if she eats the scary foods with good love and complements. Say things like " Mom needs you to eat your beans" If she continues to complain. Stand firm and explain why she needs to eat them. She is a big girl and needs to be health and beautiful! Then when she does eat, complement her! "WOW! I am so proud of you". Rewards big or small always work better then punishment.
Read about "Positive Reinforcement" in Psychology.
Good Luck!
PS If she REALLY thinks she is sick bring her to a doctor, even for your own piece of mind. Then allow the doctor to explain to her she is o.k and needs to stop complaining and also why she needs to stop complaining. Its like cry wolf, when there really is something wrong no one will take her seriously. Don't let those words come from you though. Let a doc. say them. It will be taken more seriously. Also stop feeding into it. After the doctors appointment drop it and make sure its dropped!
The point is. If she feels NOT eating food is going to get her what she wants she will keep doing it for a long time. Reward her positive eating actions With Thank Yous, I Love Yous and Good jobs!
I am guessing you are a single parent. Things like this can be hard to deal with alone.
I wish the best for you.
What she has is called Emetophobia. It is a phobia stemming from a childhood negative experience with vomiting. It can evolve into something very serious if it is not treated. I would take her to her pediatrician and ask for a recommendation for a therapist. It can evolve into anything from agoraphobia to anorexia because they are afraid to eat at all as it progresses. You will find information on the net if you do a search. Sorry- and good luck.