Vomiting can bring about the intra-abdominal pressure significantly higher, resulting in the bleeding of the capillaries of the eyes. This phenomenon can cause the rupture of small blood vessels under the conjunctiva, thus the performance of subconjunctival hemorrhage will lead to red eyes. You should remember not to rub your eyes and make the prevention of infection. The red in your eyes will be generally self-absorbed which may just need the longer time, you actually do not need to take any drugs.
Well, it seems that you know little about these problems. So, as a matter of fact, red eyes is one of the infections with eyes, and vomiting indeed could make the eyes appear to be red. And of course, vomiting could lead to red eyes, because vomiting could lead to some breakage of tiny blood vessels on your eyes. Maybe your boyfriend's problem is due to too much drinks and vomiting, it might be ok after a few days, not very serious.