In the vast majority of cases, untreated cuts do not cause gangrene. Gangrene occurs when there is disruption of blood flow to a tissue for a prolonged period of time causing death of that tissue. For a cut to cause this, it must be severe enough that a major blood vessel was compromised and was no longer able to deliver blood to the surrounding tissue. While I cannot tell you for sure if your hand is fine (since I can't see it), but I can tell you that the fact that it healed okay is very reassuring.
The most common cause of gangrene today is diabetes. Long standing diabetes can has the ability to destroy small blood vessels in the feet and hands. When injuries such as cuts or ulcers occur in these areas, there many not be enough blood flow to heal the wounds. In these cases, gangrene can result in the need for amputation of the effected area.
Since you are concerned about the wound, you should schedule an appointment with a primary care physician
. He or she can examine the wound and make sure that no additional wound care
is necessary. Hopefully, your cut will continue to heal and leave a minimal scare without impairing the function of your hand.