Can red spots occur around eyes after vomiting when pregnant?

Yes: If on the "white" part of eye these may be simple subconjunctival hemorrhages...They will clear with use of artificial tears in 7-10 days if not severe.

Yes.: Petechiae around the face and eyes are known to occur in case of severe retching and vomiting. They indicate the force of valsalva with such violent episodes and are not a health risk per se. Try reglan, phenergan, (promethazine) zofran in that order to get rid of the violent comiting episodes - and talk to your obstetrician for advice.