No.
With the implementation of the new medical system, allowlisted doctor characters will have the ability to see exactly which region of your body was injured. Even without this, the gunshot wound will show right where you were hit to anyone who is present. There's lots of ways to get shot in the head or neck and not die. People do it every day! Your doctor will collaborate with you to guide you toward the outcome you deem appropriate.
Although doctor's will also be given indication of the severity of your injury,  the seriousness of the wound is your decision as always. We highly recommend reviewing RDR Server Rule 2 regarding value of life to inform your decisions on injury severity and treatment outcome.