On Wednesday the hashtag was trending in Sydney.
As Twitter made fun of the announcement, opposition leader Bill Shorten accused Mr Morrison of having a "bizarre Captain Cook fetish".
"Mr Morrison has come to Queensland to get his historical facts wrong," Mr Shorten said.
The replica Endeavour will set sail from Sydney in 2020, stopping at 39 coastal locations and circling the country over 14 months.
Mr Morrison said the circumnavigation, which will mark 250 years since Captain Cook's voyage to Australia, would offer a new generation insight into the journey.
"As the 250th anniversary nears we want to help Australians better understand Captain Cook's historic voyage and its legacy for exploration, science and reconciliation," he said on Tuesday.
"From Far North Queensland and the Cooktown 2020 Festival across to Bunbury and down to Hobart, our government will ensure Australians young and old can see first-hand the legacy of Captain Cook and the voyage of the Endeavour."
Alongside the circumnavigation, the Cooktown 2020 Festival will receive $5.45 million to develop sites and celebrate local Indigenous culture.
Mr Morrison has since come under fire from Indigenous groups for the $6.7 million spend. They argue that the money would be better spent on Indigenous services and education.
Aboriginal Legal Services pointed out that in the same year as the re-enactment, they were facing a budget cut.
Meanwhile, co-chair of Naidoc Week and host of The Point on NITV John Paul Janke tweeted that instead of celebrating Captain Cook, we should celebrate King Bungaree - the first Australian to circumnavigate the continent alongside the British Captain Flinders.
"There are no statues of Bungaree in this country," he wrote in a message posted on Twitter.