But the National Congress for Australia's First Peoples believes he should be prepared to consult and participate with indigenous people and in their communities.
"Given that many Australians are ignorant about Aboriginal and Torres Strait Islander people, we need a prime minister who is very familiar with it," co-chair Les Malezer told reporters in Canberra.
The Congress revealed on Thursday indigenous leaders had been promised a meeting with Mr Turnbull "as soon as possible" and laid out their hopes for the future of indigenous affairs.
Mr Turnbull toppled the self-declared prime minister for indigenous Australians, Mr Abbott, on Monday night and will announce his reshuffled ministry next week.
It's unclear if Nationals senator Nigel Scullion will retain the portfolio.
Mr Malezer warned indigenous people did not need a "chief protector" in their minister and urged a different approach to the portfolio.
"It's not the white fella looking after the black fellas," he said.
"This is about having leadership and representation." Following his election as prime minister in 2013, Mr Abbott moved indigenous affairs into his department.
The Congress calls that an "absolute disaster" and says the government's new funding strategy for services left nearly every indigenous community and organisation in "some form of frustration".
Indigenous leaders have been working with the government and Labor to map a way forward on constitutional recognition, which will now need to be reconfirmed. Congress co-chair Kirstie Parker said momentum slipped with every week that passed but she believes it's too early to say if a goal of a referendum in 2017 is too ambitious.
The future of Mr Abbott's hand-picked indigenous advisory council is also uncertain and Congress would like to see it replaced with elected representatives.
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