I have red spots around my eyes from vomiting when will they go away?

1-2 weeks. The small spots (petechiae) are from burst blood vessels caused by temporarily increased blood pressure in the eye. They will take up to two weeks to disappear, and the eyes may develop a yellowish tinge for a few weeks - similar to a bruise.
Few days to a week. These are tiny bleeds. If you just have few small ones in the vicinity of the eyes will likely disappear in a few days to a week. I assume you have no other symptoms. If it worsens, consult your doc. Good luck.

Related Questions

I also have these red spots around my eyes from vomiting. Is there anything I can do to make them go away quicker -- like a cream or ointment? Thx.

No. You can use makeup concealer to cover them up but your description sounds like localized bleeding in the skin due to vomiting. These should resolve in 1-2 weeks.
Arnica gel. They should go away over a couple of weeks without treatment. You could try arnica gel which may hasten the resolution.

I've gotten red spots under my eyes. I noticed them yesterday, but I threw up today. I didn't cry or vomit yesterday. How do I get rid of them?

Time........... The red spots under your eyes probably occurred as a result of the episode of vomiting. These are small blood vessels that have ruptured and will heal--as any bruise does over the next few days to week or two.

Been throwing up for the last nine hours and now I have red spots around my eyes. What are they?

Spots around eyes. They are caused by rupture of minute blood vessels following vomiting or coughing.
Petechiae. It sounds like they may be something called petechiae. They are very small bruises, caused by a trauma such as hard coughing or vomiting. In this case they are commonly seen around the eyes. They usually go away after several days. If they appear elsewhere on the body, get bigger, or are associated with a significant illness, then you should see your doctor.