Al-Hashemi said in a statement the Iraqi Islamic Party, a major Sunni party which he heads, "... demands that the US administration deal firmly with this desecration and also calls on our government to have a position in keeping with the enormity of this humiliation".
'Heinous crime'
The US army said the staff sergeant, who fired bullets into the Quran and wrote graffiti inside it, had already been removed from Iraq and is to be disciplined.
The Association of Muslim Scholars in Iraq said in a statement: "This heinous crime shows the hatred that the leaders and the members of the occupying force have against the Koran and the [Muslim] people.
The association said it held both the US military and Iraqi government responsible for the incident.
US military authorities in Iraq have apologised to the local community in Radhwaniya, west of Baghdad, where the soldier fired into the Quran on May 11.
Colonel Bill Buckner, a US military spokesman, said the military viewed the incident "as both serious and deeply troubling," but said it was an "isolated incident and a result of one soldier's actions".
Jeffrey Hammond, a major general and head of US forces in the capital, also visited community leaders Radhwaniya to express his concern.
There was no immediate reaction from Nuri al-Maliki, Iraq's prime minister.