No. Christmas Island was named by Captain William Mynors of the British East India Company ship, the Royal Mary, when he arrived on Christmas Day, in December of 1643.
John Thompson
There are three Cook Islands, all three were named after Captain James Cook after the found them
The Cook Islands are renamed from the original Rao To Nga discovered and inhabited by Maori people
There are 15 islands (2 uninhabited. North to south, they are... Penrhyn (aka Tongareva), Rakahanga, Manihiki, Pukapuka, Nassau, Suwarrow (Suvarov) , Palmerston, Aitutaki, Manuae, Takutea, Mitiaro,...