"The crime of burning ballot box storage warehouses in the Russafa area is a deliberate act, a planned crime, aimed at hiding instances of fraud and manipulation of votes, lying to the Iraqi people and changing their will and choices".
The blaze was eventually brought under control, and confined to one of the four warehouses storing voting equipment, an interior ministry official told Reuters.
Though the cause of the fire was not reported, Mr Abadi described it as a "plot to harm the [Iraqi] nation and its democracy".
"Election material, including maybe ballot boxes, were burnt, but most of the ballot boxes were stored in another building and have been preserved", Interior Ministry spokesman Gen Saad Maan said on Sunday.
However, a Baghdad Province council member said that "all the boxes and papers have burned".
Iraq's election, held on 12 May, was won by a coalition led by populist Shiite cleric Moqtada al-Sadr, in a surprise result that pushed out political establishment figures.
Officials have suggested that most of the ballot boxes were saved from the fire, though the true extent of the damage remains to be seen.
Mr Al Abadi's statement comes as Iraq's United Nations envoy has called for an investigation into allegations of fraud and voter rigging.
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Last week, Iraq's parliament voted to dismiss the commissioners and replace them with judges while calling for a full hand recount of the approximately 11 million votes.
The Coalition referred to unclear measures taken by the Independent High Electoral Commission (IHEC) during the electronic voting as the people were used to manual voting in previous elections.
The parliament passed amendments to the election law last Wednesday, invalidating the results from overseas voters and most internally displaced votes.
There were fears that the destruction of ballots further risks the legitimacy of last month's election, which saw a major shift in Iraq's political order. State television said ballot boxes were moved to another location under heavy security.
Mr al-Sadr, who emerged as a kingmaker in the elections, has a chequered and sometimes violent past which has included targeting both foreign troops and Sunni Iraqis with violence.
"Is it not time to stand as one for building and reconstruction instead of burning ballet boxes or repeating elections just for one seat or two?", Sadr wrote.
"We have checked our election device provided to Iraq after the fraud allegation erupted, and found out that there have been no malfunction in the device nor its system", said a spokesman.
Certain parties are trying to drag Iraq into civil war, Sadr said, adding that he would not participate in one.
Opponents of the recount, mostly those whose blocs did well in the election, point out that many who voted for it were lawmakers who lost their seat.