In the same speech, Mr Erdogan called on New Zealand to restore the death penalty for the gunman who killed 50 people at two Christchurch mosques, warning that Turkey would make the attacker pay for his act if New Zealand did not.
Nearly 8700 Australians and about 2800 New Zealanders died in the Gallipoli campaign.
"We speak once again from Canakkale after 104 years, and say we have received your message", Recep Tayyip Erdogan said during ceremonies marking the 104th anniversary of the Battle of Canakkale, commemorating the martyrs in the historic victory. "Have no doubt we will send you back like your grandfathers".
"If such abhorrent views go unchecked in the media, and indeed within mainstream political discourse and institutions, minority communities of all descriptions will continue to face increasing threats of persecution around the world", they said in their open letter.
Peters rebuked the use of that footage and Erdogan's comments about New Zealand, saying he had told Turkey's Foreign Minister that it was unfair and unsafe to blame New Zealand.
At the rally, ahead of the March 31 local elections, Erdogan claimed that the manifesto by the attacker Brenton Tarrant "targeted Turkey and me", according to Bloomberg.
The New Zealand Embassy in Ankara also said Peters had raised the issue with a Turkish delegation in New Zealand.
An interfaith vigil will be held at the Paola mosque on Wednesday to honour the victims of the Christchurch attacks in New Zealand.
Deputy Prime Minister Winston Peters protested that such politicization of the massacre "imperils the future and safety of the New Zealand people and our people overseas, and it's totally unfair".
"We did not start or bring about this disaster and they clearly understood that". He made the comments in Canakkale province, northwestern Turkey which is home to the historic battlefields, on the anniversary of a World War I Turkish naval victory.
New Zealand Foreign Minister Winston Peters said on Monday that he told his Turkish counterpart and Erdogan's vice president, who were visiting the country, that the video does not represent New Zealand.
At Erdogan's campaign rallies in several cities, a video montage played blurred clips of the shooting, portions of the manifesto and remarks by far-right Australian Sen.
Erdogan said the suspect had twice visitedTurkeyand had warned that Turks have no place in Europe. Turkish authorities had begun investigating everything from hotel records to camera footage to try to ascertain the reason for his visits, the source said.