"One of the countries in the south of the Persian Gulf took care of their financial, weaponry and political needs", said President Rouhani.
"All of those small mercenary countries that we see in this region are backed by America".
"Let those who committed this horrendous crime know: the sons of Islamic Iran will not be silent in the face of these criminal acts", Alavi said.
Four assailants fired on a viewing stand in Ahvaz where Iranian officials had gathered to watch an annual event marking the start of the Islamic Republic's 1980-88 war with Iraq.
Another group, the Arab Struggle Movement for the Liberation of Al-Ahwaz, was also accused of perpetrating the attack, but a spokesman, Yacoub Hor Al-Tustari, told CNN his group was not to blame.
"It is not acceptable that these groups are not listed as terrorist organizations by the European Union as long as they have not carried out a terrorist attack in Europe", Qasemi was quoted as saying by IRNA news agency.
Thousands of people packed the streets of the southwestern Iranian city of Ahvaz to mourn the victims.
Little is known about the al-Ahvaziya group, but Islamic Revolutionary Guards Corps (IRGC) spokesperson Ramezan Sharif told ISNA it was "funded by Saudi Arabia".
"It is expected that (the Danish and Dutch) governments hand over the perpetrators of this attack and anyone related to them to Iran for a fair trial", he added.
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The former NY mayor spoke to members and supporters of the National Council of Resistance of Iran, the biggest opposition group to Iran's Islamic regime.
The hard-line daily Kayhan warned that Iranians would demand Saudi Arabia feel the "hard slap" of the country's power. "You have the Saudi Crown Prince saying that he's going to take the fight inside of Iran, you have the advisor to the UAE Crown Prince today on Twitter saying that this was not a terrorist attack, that this was declared policy.mindful of that picture, it is not completely inconceivable that Iranians are going to be looking to Saudi Arabia and potentially the U.S.as being behind these attacks".
"Iran's answer [to this attack] is forthcoming within the framework of law and our national interests", said Rouhani, adding that the United States would regret its "aggressiveness".
Elsewhere, Rouhani said that the USA seeks to cause chaos in Iran, however, he said Washington will never reach its objectives.
Saturday's attack killed 25 people, including 12 soldiers, civilians watching the parade, and a four-year-old girl.
The Islamic State (IS) jihadist group has claimed responsibility for the rare assault, which saw gunmen spray a crowd with gunfire in the southwestern city of Ahvaz on Saturday. In a statement issued on September 23, the powerful IRGC vowed "deadly and unforgettable vengeance in the near future".
Rouhani's allegation will likely ratchet up tensions with Iran's rival Saudi Arabia and its Gulf allies, which along with the U.S. have been working to isolate the Islamic Republic.
Not everyone in Iran thought it would be good idea for Rouhani to travel to the U.S.in the first place. The attack is meant to undermine Iran's security and stability and as "punishment for ... its rejection of American dependency politics", the statement claimed.
Iranian armed forces members march in a military parade similar to that struck by gunmen on Saturday.
"Rouhani will use the terrorist attack to justify Iran's presence in the Middle East".