The pup, named Puck, was rescued by medics from the British Divers Marine Life Rescue charity and is now being looked after by Emily Mayman in their short-stay recovery unit.
Puck was one of two seals injured in dog attacks in the Kilnsea area on the same day, though the others pup was not seriously hurt and has since been released into the wild.
In both incidents the owners of the dogs involved were reluctant to call their pets off and left the scene without apologising or leaving their contact details.
One man whose border collie attacked the less severely injured pup was 'reluctant' to call it to heel and another member of the public had to separate the animals.
Miss Mayman said: "Yesterday our out of hours co-ordinator received a call about an injured pup at Spurn where they're often found, as some medics have been regularly monitoring a few seals down there in the last week. The closest medics were asked to check out one particular pup after reports of a dog and public harassing it and the pup had 'bleeding injuries'.
"The medics arrived and found the pup with numerous new wounds and old scrapes. Puck had been a healthy first year juvenile grey seal only days before.
"The seal was uplifted by BDMLR along with the help of several Yorkshire Wildlife Trust members who helped transport the pup back to the rescue vehicles. It was then transported for urgent emergency veterinary treatment, where the stressed pup was sedated and had the extensive puncture wounds treated. It's now on medication to help it heal. Puck is sore and traumatised.
"There was another pup nearby which had a collie dog harassing it and physically trying to attack the pup. The owner eventually reluctantly made the dog leave it after an onlooker had to intervene to remove the dog from the seal. Unfortunately they left before details could be obtained.
"The owners of Puck's attacker also left before they could be spoken to.
"If you see a stranded pup on the beach, please give it double the amount if space you may think is necessary and call BDMLR on 01825765546 for advice. They will send a trained medic to assess the animal. Keep all dogs,children and other people away and allow it space. Never chase them back into water,"
Most of Spurn Point is a designated nature reserve, and there are some areas which are off-limits to dogs.