Two thousand members and supporters of Iraq's powerful Shiite militias demonstrated Wednesday against Saudi Arabia's execution of a prominent Shiite cleric which sparked a regional row.

The protest was staged in central Baghdad at the same time that government officials were attending military parades for Army Day, which is a national holiday in Shiite-majority Iraq.

The militiamen were also in their best uniform, carrying flags and banners bearing the portrait of executed cleric Nimr al-Nimr.

Most of the big groups in the Hashed al-Shaabi (Popular Mobilisation) paramilitary force were represented, including Ketaeb Hezbollah (Brigades of the Party of God), the Badr Organisation and Asaib Ahl al-Haq (League of the Righteous).

"The boot of a Hashed is worth more than Saudi Arabia," chanted the crowd which gathered on Tahrir (liberation) square.

"Many people say that Sheikh al-Nimr is a Saudi matter... and that Iraqis should mind their own business," said one protester, Mohammed al-Mandalawi.

"But when it comes to religion, there are no borders," he said.

The execution of Nimr on Saturday sparked outrage across the Shiite world and beyond. The Saudi embassy in Tehran was firebombed and the kingdom has since broken off ties with Iran.

Earlier protests over Nimr's execution called on the Iraqi government to respond by closing down the newly reopened Saudi embassy in Baghdad.

"Our demands to the Iraqi government are clear," said Maytham al-Allaq, a leader of the Waad Allah (Promise of God) militia.

"They include the expulsion of the Saudi ambassador from Iraq and return of the Iraqi ambassador from Riyadh," he said.

The Saudi ambassador, Thamer al-Sabhan, arrived in Iraq last week. The embassy had reopened days earlier, a quarter of a century after diplomatic relations were severed over Iraq's invasion of Kuwait.

Sabhan said that the Iraqi authorities were protecting the embassy. Prime Minister Haider al-Abadi's government has not so far hinted at any move towards breaking off ties with Riyadh.

One Iraqi official pointed out that during eight years of war between 1980 and 1988, relations between Iraq and Iran were never severed and their embassies never closed down.