President Jalal Talabani's decision sets the stage for a possible battle over the fate of the man known as the international face of the dictator's regime.
The move to prosecute and execute members of Saddam's regime is a source of controversy in Iraq, where many members of the country's Shi'ite majority, who suffered under the ousted Sunni-dominated regime, want vengeance for past crimes.
Aziz was Christian and not Sunni, but many in the Sunni community view his conviction and those of others as proof they will always be held responsible for actions carried out years ago. The Vatican has urged Iraq to commute the death sentence and said it may try to intervene diplomatically to halt it.
President Talabani yesterday cited a number of reasons for refusing to approve the execution.
"I cannot sign an order of this kind because I am a socialist," Mr Talabani said. "I feel compassion for Tariq Aziz because he is a Christian, an Iraqi Christian."
"In addition, he is an elderly man - aged over 70 - and this is why I will never sign this order," Mr Talabani said. He was speaking in Paris, where he attended a meeting of the Socialist International this week.
However, it was not immediately clear whether Mr Talabani's opposition would spare Aziz's life. Under the constitution, the president is supposed to ratify death sentences, but there are mechanisms for the execution to be ordered by parliament.
Justice ministry spokesman Abdul-Sattar Bayrkdar said death penalties can be carried out regardless of the president's refusal to sign an execution order.
"If the president refuses to sign an execution that is not a veto on a verdict," he said.