A woman locked herself to a gate outside the city's port on Friday in a bid to stop Tamboran Resources moving the rig to the drill site.
In a video posted online, Kat McNamara said she would prefer not to take such action to stop fracking in the Territory.
"I'm here because I feel morally obliged to be here," she said.
"If we don't act now, we are going to find ourselves in a situation that we can't come back from and we're going to wish people had done things like this.
"We know that the community stands with us, the science stands with us."
However, the GetUp organisation said four trucks carrying the rig had since left the port.
About 500 kilometres southeast of Darwin, the Beetaloo Basin contains an estimated 500 trillion cubic feet of gas, making it one of Australia's most energy resource-rich areas.
While fracking for gas in Australia has been used in coal seams in the past two decades, shale fracking - which is done much deeper below the surface – is yet to be trialled.
After a recent Senate inquiry into activities in the basin, Tamboran said it was pleased its "activities in the NT have been shown to be beyond reproach".
In May the NT government also lifted its five-year moratorium on fracking.
But environmental, Indigenous and some farming groups remain concerned about the impacts of unconventional mining operations, particularly on groundwater reserves.
Tamboran had no comment in relation to Friday's protest or the movement of the rig.
It previously welcomed its arrival, saying it was expected to deliver significant improvements in drilling efficiency and horsepower.
"Once operational, this onshore drilling rig will be the most powerful rig in Australia, capable of drilling more than 3000m horizontal sections," managing director and chief executive Joel Riddle said.
NT police was at the port protest but said no arrests were made.
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