Rangers injected a lethal dose of anaesthetic from an inflatable boat which took around 10 minutes to take effect. The whale was still alive when it was dragged to a nearby beach and covered with a tarpaulin.

"Everyone is very connected to this animal and it is a very emotional decision," state Parks Service chief Sally Barnes said.

"It's a bit like if you have got a family pet, when the vet says to you 'there is nothing more we can do', and the animal is suffering. I've taken the decision that I don't want to prolong the agony of this animal," she said.

A vet report and blood tests on Thursday revealed the two-tonne calf, believed to be only two to three weeks old, was in poor condition and had only hours to live. It was suffering from shark bite wounds and breathing difficulties.

A team of park rangers and marine scientists surrounded the animal in boats and decided to put it down shortly after sunrise.

An Aboriginal "whale whisperer" was brought to the bay on Thursday to "talk" to the calf. Colin responded by lifting its head, whisperer Bunna Lawrie recounted afterwards.

"He heard me singing and came over. I looked at him and he was full of life. He had a few scratches and cuts on him and I was a bit worried about his eyes. All he needed is a mother's milk," Lawrie told the Sydney Morning Herald newspaper.

He pleaded with authorities for a change of heart.

The whale's struggle to survive captivated Australians, who strongly oppose Japanese "scientific" whale killing and flock to whale-watching tours during the giant mammals' annual migration to the Antarctic and return to breed in warmer Australian waters.

On Monday a team of workers towed the private yacht out to sea to try to lure the calf into deeper water, hoping that it would find its mother or another passing whale pod, but it was spotted close to the beach at Pittwater again on Tuesday.