How do doctors treat a broken blood vessel?

Usually nothing: This depends on the cause of the "broken blood vessel". Loss of supporting tissue associated with "maturity", medications like Aspirin or non-steroidals, varicosities or local trauma are common causes. The "problem" is usually more cosmetic rather than clinical unless there's blood where it shouldn't be; the body usually takes care of this gradual absorption of the blood like a healing bruise.

Size matters: Depends on the injury: it can be done by application of pressure, sutures or in some cases ligature or graft.

Many answers: There are a variety of different techniques used to fix vessels that are damaged. It depends on the vessel and cause of damage. Some vessels can be repaired by removing a small area of damage. Other repairs require replacement with the patient's own vein, which would be taken from another site on the body. Synthetic tubes are used for repairs and can be inserted by endovascular techniques.