The embassy released a statement on Sunday afternoon saying the country wanted to "categorically rule out" the existence of a security threat against New Zealanders.
Peters told reporters he had not addressed these comments with Erdogan because he did not think it would serve a "long-term peaceful objective", adding that he had received assurances regarding the safety of New Zealanders visiting Turkey.
Turkish President Recep Tayyip Erdogan shakes hands with Foreign Minister Winston Peters during an Organization of Islamic Cooperation meeting.
No other heads of state or government attended the gathering of the OIC, which groups together Muslim countries to protect the interests of the Muslim world.
Mr Peters took a conciliatory tone on Friday, welcoming comments by Turkish foreign minister Mevlut Cavusoglu, who said at a news conference at the end of the OIC meeting that Australians and New Zealanders visiting Turkey would be greeted at Gallipoli remembrance ceremonies next month with the same welcoming hospitality "as they always were".
"This person will face the full force of New Zealand law and spend the rest of his life in isolation in a New Zealand prison", he said.
A diplomatic dispute was triggered between the two nations after President Erdogan showed the footage of the horrific New Zealand shooting at a series of election rallies last week, including on Thursday.
Hundreds of New Zealanders and Australians travel to Canakkale each year for Anzac Day on April 25, to mark the start of the battle of Gallipoli and to commemorate the dead.
Erdoğan then referred to the Anzac conflict in World War I and said: "We will see you off like your ancestors".
Peters later told reporters he did not ask Erdogan to stop showing the video of the shooting in Christchurch at election rallies because he understood its screening had stopped.
He has also shown extracts from a "manifesto" posted by the attacker and later taken down, drawing condemnation from New Zealand and Australia.
The minister said he had been reassured by Istanbul that New Zealand citizens would be welcome in Turkey.
Thousands of people gathered in New Zealand for Friday prayers and to pay their respects during a mass funeral.
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