Mr Andhoy set sail from Auckland last month on the ship Nilaya, aiming to retrace the steps of a previous expedition that made headlines last year when three support crew members died.
He and companion Samuel Massie had attempted to cross the South Pole on quadbikes but the trip turned to disaster when the waiting support yacht Berserk sank in McMurdo Sound.
Mr Andhoy failed to get permission to undertake both expeditions.
Mr Morgan, who is currently on a trip to Antarctica to raise awareness of the importance of the sub-Antarctic islands, has slammed the Norwegian's latest mission.
He has described Mr Andhoy as "callous'' and said in a blog earlier this week he hoped he sunk.
"He appears to not give a toss about the amount of hurt he imparts on those who get in the way of his quest for his 'Wild Vikings' brand to attract sponsors and book sales.
"It's pathetic, really, and one has to have genuine sympathy for the families of the three lost souls that his quest for fame condemned to death.''
Further to his blog, Mr Morgan has now called on the New Zealand Government to implore the Norwegian Government to act.
In Norway Mr Andhoy is often conveyed as a hero, but Mr Morgan said Mr Andhoy's trip presented risks to the environment, other people and ships, and said it was a direct result of negligence from the Norwegian Government.
"It is about time the New Zealand Government made representations to Norway and implore it to behave as a member of the Antarctic Treaty.''
The New Zealand Government needed to lodge the strongest protest it could about Norwegian "lethargy'', Mr Morgan said.
"Imagine if as a result of Norway's slovenliness, if we get 20 Andhoy wannabes clowning around down here showboating their disregard of environment and shipping.''
Mr Morgan expected to cross paths with Mr Andoy in Antarctica as they were both heading to the same place, Backhouse Bay.
On Wednesday Norwegian media reported Mr Andhoy's ship had arrived in Victoria Land, north of New Zealand's Scott Base, where they began a search for debris from the Berserk.
The Nilaya then set course for Franklin Island, an agreed emergency rendezvous point on the fateful Berserk mission.
Also on board the ship is Aucklander Busby Noble, a friend of the men who died, who reportedly planted a Mana Party flag in the polar ice.
