Recep Tayyip Erdogan said Turkish investigators had evidence the Saudi writer was the victim of a premeditated “savage murder” by a 15-man hit-squad.
His version of events based on what he described as “new evidence and information” contradicts Saudi Arabia’s explanation that Mr Khashoggi had died in a “fist fight” at the Saudi consulate in Istanbul on 2 October.
Mr Khashoggi was a known critic of the Saudi government and the country’s crown prince, Mohammed bin Salman.
The Turkish president did not mention the crown prince in his speech, though officials linked to the royal have been implicated in the killing.
Saudi Arabia has said the heir-apparent of the world’s top oil exporter was not involved, but any major decision must be signed off by the highest powers within its ruling Al Saud family.
Mr Erdogan said Riyadh’s suggestion it was a rogue operation “will not satisfy us” and said there should be no diplomatic immunity for his killers.
He said 18 people had been arrested – three from the consulate and a further 15 who are believed to be part of a hit squad who flew to Istanbul to carry out the killing.
“To blame such an incident on a handful of security and intelligence members would not satisfy us or the international community,” Mr Erdogan said in an address to the Turkish parliament.
“Saudi Arabia has taken an important step by admitting the murder.
“As of now we expect of them to openly bring to light those responsible – from the highest ranked to the lowest – and to bring them to justice,” he said.
Mr Erdogan gave no specific details on how 59-year-old Mr Khashoggi was killed.
There was also no mention of an alleged audio recording that Turkish authorities claim to have of Mr Khashoggi’s death that supposedly confirms he was tortured, killed and was dismembered.
The Turkish president demanded answers from Saudi Arabia on who gave the orders for the killing and the whereabouts of the journalist’s remains.
Reporting from Istanbul, Sky’s foreign affairs editor Deborah Haynes said Mr Erdogan’s promise of the “naked truth” about the killing “left more questions than answers”.