Yuriy Lutsenko said any investigation into Mr Biden and his son would have to start in the US.
"I don't know any reason to investigate Joe Biden or Hunter Biden according to Ukrainian law," he said.
Mr Trump's efforts to have Ukraine investigate the pair prompted an impeachment inquiry by the Democrats.
A transcript of a call Mr Trump made to Ukraine's new President Volodymyr Zelensky on 25 July shows he urged him to investigate discredited corruption allegations against Mr Biden and his son.
Mr Trump and his allies have been suggesting that Mr Biden, as Barack Obama's vice-president, encouraged the firing of Ukraine's top prosecutor, Viktor Shokin, because he had been investigating Ukrainian natural gas company Burisma, which employed Hunter Biden.
Hunter Biden became a director at Burisma in 2014, while his father held a key role in US policy towards Ukraine.
Mr Biden is currently frontrunner to be the Democrat to take on Mr Trump in the November 2020 presidential election.
What did Lutsenko say?
Speaking to BBC Kiev correspondent Jonah Fisher, Mr Lutsenko – who succeeded Mr Shokin and stood down last month – said there was no plan to open the investigation into Burisma, and that any investigation into Hunter Biden would have to start in the US.
"It is the jurisdiction of the US," he said, adding that any "possible embezzlement" at Burisma "happened two or three years before Hunter Biden became a member of the board".
Mr Lutsenko also said he had met Donald Trump's personal lawyer Rudy Giuliani, who asked him if it was possible to investigate the Bidens in his country.
"I told him the same I told you – it is not my jurisdiction," he said. "It is likely to be the jurisdiction of the US. If you will send me a request, yes, I will give you all official information, but it is not Ukrainian jurisdiction – that was my answer."
Pressed by our correspondent on any evidence Ukraine may have against the Bidens, Mr Lutsenko said: "I can do nothing [on a case] which is not connected with Ukrainian law."
What is the impeachment inquiry about?
In a complaint filed last month, a whistleblower from the intelligence community reported his "urgent concern" that the president had used his office to "solicit interference from a foreign country" in the 2020 US election.
The impeachment inquiry launched on Tuesday now focuses specifically on Mr Trump's call to the Ukrainian president about the discredited Biden allegations.
Democrats accuse Mr Trump of illegally seeking foreign help in the hope of smearing Mr Biden.
Mr Trump is also accused of using military aid for Ukraine as a bargaining tool.
President Trump has dismissed the proceedings as a "hoax" and "another witch-hunt". On Saturday he accused leading Democrats of "presidential harassment" and branded them "Democrat Savages".
