Well, of course, it is not a sign of pink eyes. And generally speaking it is normal to have bloodshot eyes for most of newborn babies. So you do not need to worry about it. And according to some experts, it can be related to the high pressure in blood. Also we can say that it's from the pressure from being squeezed down the birth canal. And of course, it will heal up on its own. Anyway, it will just be fine.
Well, it may be so serious. Newborn babies are commonly weak and some organs still do not develop completely. Thus, your newborn baby began to occur bloodshot in eyes showed that there is something wrong with baby eyes. As for adults, bloodshot in eyes may due to pink eye, black eyes and eye irritation, even allergic. While for newborn babies, the most possible factors are cold, infection and blood vessels in eyes have some problems. If your baby just has a little bloodshot in eyes, you can take easy in that this bloodshot may cause by bacteria and virtues enter eyes. At this time, you just need to see a doctor and take some notes from him/her. If not, this condition may be serious healed. Blood vessels in eyes are not so strong and blurry vision or other eye problems can be caused by blood vessels disorder. You must let your baby have a thorough eye exam to determine the real reasons. No any self-care before visiting a doctor.
Actually, bloodshot eye is very common in newborn babies. When the baby is around the corner to the world, its head undergoes a tremendous amount of pressure. Tiny blood vessels in the sclera can burst from the pressure. It is a bruise on the eyeball. However, it will take a few weeks to go away as a rule, and in most time, there is no permanent damage to the eyes. So, you don't need to worry so much.