As has been mentioned, "being sick" is often used as a polite euphemism for "vomiting", and is widely used for that purpose.
I suspect that your friend was referring to a person being ill (feeling miserable, suffering a disease, etc), and in that case he or she was correct. Ordinarily you would say "He is sick" rather than "He is being sick".
Generally, the construction "He is being (something)" is used when (something) is a matter of choice, such as "He is being greedy", or "He is being obstinate" or "He is being generous", while being sick is not a matter of choice.
I can imagine an exception, but it is considerably involved. Let's say a mutual friend is well-known for his hypochondria, and has claimed it repeatedly when it seems to conveniently remove him from some obligation. You might say to another friend, "Don't expect to see him today. He has a paper due, and he hasn't finished it, so he's being sick." Here, "being sick" is used as a sarcastic replacement for "malingering".