Here is an emergency room video of a relatively minor nail gun injury. The nail did not hit an artery and if the fellow is lucky he will not get an infection or suffer any nerve damage. At the very least, though - he's going to be in pain, miss work, and have to pay some medical bills.
If you watch the video you'll understand why they tell you to seek medical treatment and not pull nails out yourself. Before pulling the nail, the doctor uses an ultra-sound device to see if the nail hit an artery. And after the nail is out he flushes the wound clean in a way that would be impossible to do at home (or without anesthetic).
It is a Doppler, not an ultrasound. You shouldn't rotate the nail with extrication since it has either a wire or glue on the shaft that could create further damage on the way out. I'm guessing he impaled the flexor tendon and this will need splinting while it heals and is at risk for infection. This is potentially a very disabling injury and is nothing to fool with.
IT SEEMS TO ME THE NAIL COULD HAVE BEEN CLIPPED OFF CLOSE TO THE POINT OF ENTRY AND PULLED THROUGH THE EXIT HOLE THUS MINIMIZING THE INJURY . ONE WOULD THEN AVOID THE RASPING ACTION OF THE BONE BY THE CIRCULAR RIDGES THAT WERE SHAPED TO THE NAIL FOR RESISTANCE TO REMOVAL