"We call upon you, with a saddened heart and sorrow because of what is happening in Iraq and because the people are suffering from violence, poverty and misery," Sako said in a statement.
His appeal followed a mass protest for political reform April 30. Thousands of demonstrators, mostly supporters of the Shiite cleric Muqtada al-Sadr, breached the walls of Baghdad's Green Zone, where the seat of the Iraqi government and most foreign embassies are located. They overtook the parliament building.
"You are all aware of the failure of government institutions, of the disruption of the law, of delayed solutions and essential reforms demanded by people," Sako said.
"All of this is a result of the divided political scene, of the failure to achieve real national reconciliation, as well as the conflict of interests and ambitions that has been demonstrated recently," he added.
"Relying on our human, national and moral responsibilities, we urge all Iraqi politicians to adopt wisdom and quietness, in addition to making every possible effort for true reconciliation to put an end to such institutional, economic and security deterioration," the patriarch said.
"We are also calling for unity of all the counterparts to have a clear vision and a shared co-ordinated plan to liberate all Iraqi territories and work together for peace and stability in the country to enable displaced people to return home."
Thousands of families, most of whom are Christian, have been displaced by Islamic State offensives in the Iraqi region of Mosul and the Ninevah Plain.
"Enough is enough! We have had enough division and dispersion," Sako said.
"Focus on the future of your country, the future of your fellow citizens," he urged the leaders. "As your brothers — millions are dying from poverty and diseases — Iraqis deserve much better than that."