Tiwi Islander Dennis Tipakalippa launched the legal action in June, claiming the offshore gas regulator should not have allowed drilling in the Barossa gas field, 265 kilometres northwest of Darwin.
The Munupi elder said he was not consulted over the company's environmental plan and feared the project could damage his people's sea country.
Federal Court judge Mordecai Bromberg in September ruled in Mr Tipakalippa's favour, finding the regulator should not have approved Santos' environmental plan.
He ordered for the approval to be set aside and drilling to be stopped, but Santos appealed the decision.
The gas company argued during a hearing last month that Justice Bromberg did not properly consider what constituted a "relevant person" who needed to be consulted.
Santos barrister Christopher Horan KC told the court that under the regulations, a relevant person was someone whose "functions, interests or activities" could be affected by the drilling work.
Mr Horan said while the Tiwi Islands' traditional owners had a connection to sea country that was genuine and real, it did not constitute the type of legal interest detailed in the legislation.
But Claire Harris KC, representing Mr Tipakalippa, argued that definition was too limited and ignored a person's social, cultural and spiritual interest in an environment.
Ms Harris told the court the Tiwi Islanders had a profound connection to their sea country, which should be understood as a relevant interest under the laws.
Federal Court judges Susan Kenny, Debra Mortimer and Michael Lee heard the appeal in Melbourne last month.
Justice Mortimer will deliver judgment on behalf of the full court on Friday afternoon.
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