Australian Brenton Tarrant, 28, a suspected white supremacist, was charged with murder on Saturday after a lone gunman opened fire at the two mosques during Friday prayers.
Erdogan — who is seeking to drum up support for his Islamist-rooted AK Party in March 31 local elections — said Turkey would make the suspected attacker pay if New Zealand did not.
The comments came at a campaign rally that included video footage of the shootings which the alleged gunman had broadcast on Facebook.
Ardern said Peters would seek urgent clarification.
"Our deputy prime minister will be confronting those comments in Turkey," Ardern told reporters in Christchurch. "He is going there to set the record straight, face-to-face."
Peters had earlier condemned the airing of footage of the shooting, which he said could endanger New Zealanders aboard.
Despite Peters' intervention, an extract from Tarrant's alleged manifesto was flashed up on a screen at Erdogan's rally again on Tuesday, along with footage of the gunman entering one of the mosques and shooting as he approached the door.
Meanwhile, Australian Prime Minister Scott Morrison said he summoned Turkey's ambassador for a meeting, during which he demanded Erdogan's comments be removed from Turkey's state broadcaster.
"I will wait to see what the response is from the Turkish government before taking further action, but I can tell you that all options are on the table," Morrison told reporters in Canberra.
Morrison said Australia's ambassador to Turkey will on Wednesday meet with the members of Erdogan's government.
Morrison said Canberra is also reconsidering its travel advice for Australians planning trips to Turkey.
Relations between Turkey, New Zealand and Australia have generally been good. Thousands of Australians and New Zealanders travel each year to Turkey for war memorial services
Just over a century ago, thousands of soldiers from the Australian and New Zealand Army Corps (ANZAC) struggled ashore on a narrow beach at Gallipoli during an ill-fated campaign that would claim more than 130,000 lives.
The area has become a site of pilgrimage for visitors who honour their nations’ fallen in graveyards halfway around the world on ANZAC Day every April 25.
Erdogan said at a campaign rally that Turkey would punish the shooter if New Zealand did not.