The group provided no evidence for the claim.
Gunmen fired on a military parade in southwestern Iran on Saturday, killing 24 people, half of them members of the Revolutionary Guards, state news agencies reported, in one of the worst attacks ever on the elite force.
While Iranian officials have not directly named the Gulf states, their comments are believed to be directed at Saudi Arabia, the UAE and Israel, which all have hostile relations with Iran and have promised to counter its influence in the region, including inside the country.
Iran has suggested that it could take military action in the Gulf to block other countries' oil exports in retaliation for USA sanctions meant to halt its sales of crude.
Haley said Washington was trying to counter Iranian malign activities in the region as Tehran continued to test ballistic missiles, support terrorism and sell arms.
But while violence and bomb attacks are not uncommon on both Iran's western and eastern frontiers - bordering Iraq and Afghanistan respectively - audacious and organized assaults that directly target the Islamic Republic's security services in city centers are still rare.
"It is absolutely clear to us who has done this, which group it is and to whom they are affiliated", Rouhani said on state television on Sunday, shortly before leaving Tehran for the UN General Assembly in NY.
"Those who have caused this catastrophe. were Saddam's mercenaries as long as he was alive and then changed masters", he said, referring to late Iraqi dictator Saddam Hussein.
He alleged that a Gulf country had supported the "financial, weaponry and political needs" of the attackers.
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"We will never decrease our defensive capabilities. we will increase them day by day", Rouhani said at a military parade.
"Those who opened fire on civilians and the armed forces have links to the Ahvazi movement", Guards spokesman Ramezan Sharif told ISNA.
"Hopefully we will overcome these sanctions with the least possible costs and make America regret its aggressiveness toward other countries, and particularly Iran", said Rouhani.
The attack lasted about 10 minutes, the BBC's David Bamford told NPR.
Iran's state-run IRNA news agency says 20 people have been wounded in an attack on military parade in Ahvaz.
A 2017 double bombing in Tehran claimed by the Sunni extremist group Islamic State left 18 people dead. A local news agency in Khuzestan province, of which Ahvaz is the capital, aired grainy mobile phone footage showing parade goers fleeing as soldiers lay flat on the ground.
A number of towns in Khuzestan saw some of the most violent clashes between protesters and security forces in the anti-government unrest that broke out late past year.
Who carried out the assault also remained in question. "The Iranian people and the armed forces will respond to this", he was quoted by IRNA as saying.
Iran summoned diplomats from Denmark, the Netherlands and Britain over their "hosting of some members of the terrorist group" which carried out the attack, state media said Sunday.
All four attackers were killed.
A Revolutionary Guards spokesman claimed the attackers were "trained and organised by two Gulf countries" and had ties to the USA and Israel. Some analysts wondered, however, why there wasn't a stronger response from the United States. In a statement issued on September 23, the powerful IRGC vowed "deadly and unforgettable vengeance in the near future".