In the absence of its ambassador, Turkey's charge d'affaires has been "invited" to respond to "the Islamic republic's objection and regret over Erdogan's inappropriate and unusual comments," foreign ministry spokeswoman Marzieh Afkham said in a statement.
"We demand a clear and convincing response," Afkham added amid calls for Erdogan's planned visit to Tehran next month to be cancelled.
Turkey has expressed support for a Saudi-led coalition that is conducting air strikes against Shiite Huthi rebels in Yemen.
Hadi and his backers charge that the rebels, who overran the capital last year and have since advanced across much of the country, have support from Iran.
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On Thursday, Erdogan demanded that "Iran and the terrorist groups must withdraw" from Yemen.
He also accused Iran of meddling in other regional countries, citing its role advising and coordinating Shiite militia groups in the fight against Islamic State (IS) jihadists in Iraq.
"The aim of Iran is to increase its influence in Iraq," said Erdogan.
The comments have been condemned by Iranian politicians and newspaper editors who demanded that the foreign ministry withdraw the invitation for Erdogan to visit.
"Erdogan's trip to Iran is an insult to our people and a betrayal of the resistance," Hussein Shariatmadari, the editor of the leading conservative newspaper, Kayhan, said yesterday.
"An immediate cancellation is the least expected from the ministry of foreign affairs."
Esmail Kosari, a member of parliament's national security and foreign affairs committee, warned: "If the foreign ministry does not cancel the president's trip, parliament will next week take up this issue."
"Iran's approach to the region and relations with neighbours is based on peace, stability and cooperation based on mutual respect," she said.
"We believe that Iran-Turkey cooperation can meet this goal.