"America is acting like a bully toward the rest of the world.and thinks it can act based on brute force", Rouhani told reporters just before leaving Iran for the United Nations meeting in NY.
The United States reimposed crippling sanctions on Iran last month, and a new round of even harsher sanctions targeting Iran's vital oil sector is set to go back into effect on 5 November.
"This crime is a continuation of the plots of the regional states that are puppets of the United States, and their goal is to create insecurity in our dear country", Khamenei said in a statement published on his website.
Mr Khamenei did not name the "regional states" that he believed were behind the attack.
"The Islamic republic will give a crushing response to the slightest threat against the country", he vowed.
The Arab Struggle Movement for the Liberation of Ahwaz is classified as a terrorist group by the Iranian government.
Iran in response summoned diplomats from Denmark, the Netherlands and Britain to complain about them "hosting some members of the terrorist group" and "double standards in fighting terrorism", the foreign affairs ministry said.
Islamic State militants also claimed responsibility. Neither claim provided evidence.
"These terrorists. were trained and organised by two".
Several armed men who had been standing in the crowds watching the parade began firing randomly, Brigadier-General Abolfazl Shekarchi, a spokesman for the IRGC, told state TV in a phone interview.
The Iraqi border crossing authority said Saturday that the Iranian side was temporarily closing al-Sheeb and al-Shalamcha border crossings between the two countries after the attack.
He was referencing the moment when at least 24 people including women and children were gunned down at a military parade in southwestern Iran on Saturday.
Ali Vaez, the director of the Iran Project at the International Crisis Group (ICG), tells RFE/RL that Tehran appears to have intelligence of renewed activities of separatist groups and their ties to some Persian Gulf countries.
Emphasizing that a number of soldiers, as well as event organizers, were also shot during the parade, he said the most victims and those badly injured are ordinary people who were watching the event, noting that "we can not accurately declare the death toll at the present time".
The incident comes at a critical time for Iran's ruling elite as they try to contain the fallout from the US decision to abandon the 2015 nuclear deal and re-impose crippling economic sanctions against the country. Images captured by state television showed journalists and onlookers turn to look toward the first shots, then the rows of marchers broke as soldiers and civilians sought cover under sustained gunfire.
But attacks on regime targets inside major cities are far rarer.
TRT World spoke to political analyst Mohammad Marandi about the attack.
Hezbollah in a statement said "satanic and malignant hands are behind this terrorist action", and accused the United States and its allies of conducting the attack "directly or by proxy".
The province has a large ethnic Arab community, many of them Sunni, the Muslim denomination to which IS adheres and which is a minority in Shiite Iran.
The semiofficial ISNA news agency said an unnamed spokesman for the IRGC had blamed Arab nationalists backed by Saudi Arabia for the attack.