This follows Australia's Prime Minister Scott Morrison issuing a strong rebuke to the Turkish President and is reportedly considering expelling Turkey's ambassador after Erdoğan's "highly offensive and highly reckless" comments.
He said Australia's Security Agencies was now reviewing whether it is safe for Australians to travel to Gallipoli for Anzac Day services.
But in a statement, the Ministry of Foreign Affairs and Trade (MFAT) said its travel advice for Kiwis travelling to Turkey had not changed.
Meanwhile, Act Leader David Seymour called Erdoğan "an idiot" and said his recent behaviour proves that.
Seymour said Peters needed to "stand up for the interests of humanity at this time and take a principled stand against what Mr Erdoğan said, which was absolutely deplorable".
After using the Christchurch mosque attacks as a platform for electioneering, Erdoğan has struck a much more conciliatory tone in an op-ed for the Washington Post today.
In the piece, Erdoğan praised Prime Minister Jacinda Ardern for her leadership.
"All Western leaders must learn from the courage, leadership and sincerity of New Zealand's Prime Minister, Jacinda Ardern, to embrace Muslims living in their respective countries."
He also challenged other Western countries, saying they have "certain responsibilities" after the terror attack on Friday.
Erdoğan called on them to reject the "normalisation of racism, xenophobia and Islamophobia, which has been on the rise in recent years".
In the past, Erdoğan wrote, xenophobia and Islamophobia have been met with "silence in Europe and other parts of the Western world".
"We cannot afford to allow this again.
"If the world wants to prevent future assaults similar to the one in New Zealand, it must start by establishing that what happened was the product of a co-ordinated smear campaign."
In the op-ed, Erdoğan stressed that there was "absolutely no difference between the murderer who killed innocent people in New Zealand and those who have carried out terrorist acts in Turkey, France, Indonesia and elsewhere."
His comments come after he struck a much less diplomatic tone when speaking about the shootings earlier in the week when he invoked the Anzacs' Gallipoli campaign during World War I.
In a message, apparently aimed at New Zealanders and Westerners, Erdoğan said: "Your grandparents came, some of them returned in coffins. If you come as well like your grandfathers, be sure that you will be gone like your grandfathers."
He added that "the enemies of Muslims have shown that they continue to hate us" and "they are testing us from 16,500km away, from New Zealand, with the messages they are giving from there. This is not an individual attack, it is organised".
Erdoğan has also shown clips from the video the gunman made of the shooting.
The Turkish President is on the campaign trail ahead of Turkey's elections later this month.
Those comments, and the showing of the video, have been criticised by Foreign Minister Winston Peters, who is travelling to Turkey this week to "set the record straight".
They also drew condemnation from Australian Prime Minister Scott Morrison, who has called in the Turkish ambassador to demand answers, according to reports.
Peters will be attending an emergency meeting of the executive committee of the Organisation of Islamic Co-operation (OIC) - a 57 state organisation representing the voices of Muslims around the world - this week to discuss the terrorist attack.
The emergency meeting was convened at the request of Turkey.
The OIC secretary-general, Yousef A. Al-Othaimeen, will present an overview of OIC's efforts to raise awareness of the threat posed by Islamophobia and countering all forms and manifestations of terrorism, as well as the efforts made to protect and support Muslim communities throughout the world.
Peters said: "They asked me to come and I'm going to go.
"I certainly intend to put New Zealand's record as being an innocent party to an act of a foreigner in our country."
