I am not a doctor or in the medical field. You should not need to see a dr. When you give blood they are "bursting" the vein with a needle to take blood. They are able to stop the bleeding in most cases by applying pressure (but not enough to cut off blood flow) to the area. A blue mark or greenish brown blue may still remain because it is just like a bruise.
So if you are worried about still bleeding then apply light pressure to the area for 5mins should stop it.