Eating properly nourishes your body and makes you (keeps you) healthy. Not eating, starving yourself, makes you sick.

I myself noticed that when I starve myself for fear of vomiting, I feel more nauseous than if I eat. Actually, when I eat well, I feel a whole heck of a lot better.

If your mind will not allow you to eat a whole meal, start by grazing... take bites of foods you otherwise would not eat. Just nibble throughout the day, introduce new foods (healthy foods you avoid) into your body this way. Your mind will soon make the connection that eating "x" will not make you sick, and you'll be able to eat more of it until you can manage a whole healthy meal.

Eating does not make you sick, being sick makes you sick. And starving yourself greatly increases the odds of you catching things things that will make you sick because a malnourished body is a weak body that can't fight off anything, as well as being malnourished makes you sick in and of itself.

Please, slowly challenge your thinking, challenge your eating habits, take ONE bite of something you know you should be eating but are afraid to... do it now. See what happens. I promise you won't get sick. You'll most likely feel anxious, but that's your fear, not the food.