Gunmen sprayed a crowd with bullets in the city of Ahvaz, killing at least 25 people, including children during the parade.
The "formal incitement against the UAE from within Iran is unfortunate, and has escalated after the Ahvaz attack", said Minister of State for Foreign Affairs Dr Anwar Mohammed Gargash in a tweet.
On Monday, thousands of mourners gathered at the Sarallah Mosque on Ahvaz's Taleghani junction, carrying caskets in the sweltering heat.
"Based on reports, this cowardly act was done by people who the Americans come to help when they are trapped in Syria and Iraq, and are paid by Saudi Arabia and the U.A.E.", Khamenei was quoted as saying on his official website.
A video released by the IS-affiliated Amaq news agency showed three men in a vehicle who appear to be dressed in IRGC uniforms. "We stand with the Iranian people against the scourge of radical Islamic terrorism and express our sympathy to them at this bad time", Nauert said.
Brig-Gen Hossein Salami said in a speech to mourners at Monday's funeral: "You have seen our revenge before. we promise that our response will be crushing and devastating and you will regret what you have done".
Explaining why Iranian athletes do not take part in any match with the representatives of the Zionist regime of Israel, Ayatollah Khamenei said, "After the victory of the Islamic Revolution, the Islamic Republic did not recognize the Zionist regime and the [former] racist regime of South Africa".
"Considering (the Guards') full knowledge about the centers of deployment of the criminal terrorists' leaders., they will face a deadly and unforgettable vengeance in the near future", the statement said.
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Senior commanders of the Islamic Revolutionary Guards Corps (IRGC) have said the Ahvaz attack was carried out by militants trained by Gulf states and Israel, and backed by America.
"The terrorists themselves have perished, our agents will identify their remnants and supporters to the last man".
But that remark came a week after Trump and Rouhani traded threats, and Iran's leadership replied by saying the time for talks was before, not after, Trump withdrew the USA from the Iran nuclear deal.
The Islamic State group also claimed responsibility.
An Iranian soldier carries an injured colleague after the attack.
What did Iranian leaders say?
Accusations against Gulf countries will nearly certainly antagonise Iran's regional foe Saudi Arabia.
On June 7, 2017 in Tehran, 17 people were killed and dozens wounded in simultaneous attacks on the parliament and on the tomb of revolutionary leader Ayatollah Ruhollah Khomeini - the first inside Iran claimed by IS.
The regional superpowers support opposing sides in the civil wars in Yemen and Syria, as well as rival political groups in Iraq and Lebanon.