Adnan al-Dulaimi, one of the leaders of the National Concord Front coalition, said the group rejected the results announced by the commission.
"If the commission does not take steps to restore justice to other lists, we will demand a new election be held," he said.
However, Abdul Husain Hindawi, the head of the commission, overruled the complaints, saying on Tuesday: "So far there are no objective grounds to order a rerun in any province."
The commission had expected such complaints after Thursday's parliamentary poll, he said.
Sunni Arabs had turned out en masse to vote in Iraq's landmark election last Thursday for the first full-term government following the fall of Saddam Hussein. They had boycotted elections for a transitional parliament in January.
Hindawi described the demand for a repeat vote as "political" and said: "No one is satisfied with the results but those who won are less critical than others, of course."
But Hindawi said the commission would study any complaint.
"Their position is rather political," he said of the Sunni complaints.
Thursday's vote was marked by
an estimated 70% turnout
"If they have proof of fraud, then they should send a letter to the commission and we will reply to them. But I'm sure that they will also be dissatisfied with our reply."
Hindawi acknowledged there may have been some violations but said they were minor and would not affect the overall result.
The largest Sunni party in Iraq earlier said results of the Baghdad vote in last week's national election were fraudulent and the electoral commission, the IECI, should order a new ballot.
Call for revision
Tariq al-Hashimi, leader of the Iraqi Islamic party, called on Tuesday for an immediate revision of the figures.
Al-Hashmi was speaking a day after the IECI issued partial results which suggested that the main Shia coalition had fared well.
Attacks and kidnappings have
resumed in Baghdad after a lull
"They should ... immediately revise the figures. The ball is now in the court of the IECI," he said.
Also on Tuesday, the US ambassador took an apparent swipe at the current Iraqi interior minister, a conservative Shia facing accusations of allowing mistreatment of Sunnis.
Iraq must have an interior minister who rejects sectarianism, Zalmay Khalilzad - seen as a key mediator in the formation of Iraq's next government - told a year-end news conference.
He said Iraqis had to cooperate across ethnic and sectarian lines if their country was to succeed.
The choice of interior minister after a new government takes office will be crucial, Khalilzad said.
He did not mention any names, but his comments came as the Interior Ministry - run by the main Shia party in power - battles accusations that it has been singling out rival Sunni Arabs for abuse and torture.
Bayan Jabor, the interior minister who is also a member of the powerful, Iran-allied Supreme Council for Islamic Revolution in Iraq, has denied the allegations.
Bayan Jabor has denied charges
of Sunni detainees' torture
In other developments on Thursday, Mahmud Saadat, 50, a driver for the Jordanian embassy in Baghdad was seized as he drove his private car from home to work, Nasir Juda, a Jordanian government spokesman, said.
Saadat's car was "intercepted by three vehicles along the way from home to work this morning", Juda said.
Yasir Abu Hilala, Aljazeera's Jordan bureau chief, said relatives of Saadat, a Jordanian, appealed to his captors to release him, saying he had nothing to do with politics or the embassy.
Saadat has 15 children, Hilala added.