It could be, but you need to see your doctor for a thorough check-up, the mind is a powerful thing. Normally with an eating disorder you just refuse to eat, and usually don't feel sick afterwards. But it's great that you recognize that something is not right and are seeking help. Rule out other causes first,there may be an underlying reason you feel sick after eating, or it may be an eating disorder, but you need to know for sure. If it is anorexia therapy can help you get beyond this and live a healthier, happier life. One step at a time, and sharing with us was a big step! Now make an appt. with your doctor and let's go from there.
yes, you can give yourself an eating disorder that way. and it's not really an effective way to lose weight either bc your body will go into starvation mode, making it difficult to shed excess lbs, plus you won't have any energy to exercise.
i've been having trouble eating too, mostly bc of meds - in the past i have had issues with anorexia and bulimia tho. now i literally force myself to eat at least three meals a day, although i usually find it easier to eat smaller meals, like 200-300 calories, five or six times a day. set a reminder on your phone if you have to. even if you don't finish the whole portion try to eat a little.
also make sure you are eating fresh fruit/veggies, raw nuts, etc - they are less likely to leave you feeling sick after. i get sick every time i try to eat processed food for some reason but what my brother calls my "hippie food" seems to work out well.
if you don't already, learn how to cook. it can make eating more fun and less psychological... i don't know just some ideas. just make sure you eat, once you get to far into an eating disorder it is really hard to get back out.