"I affirm that the step which we have carried out was a reaction to a group of actions that caused paralysis of the democratic institutions in the country,” he said.
Abdel Aziz also said that the state council, which he heads, would soon hold presidential elections.
Military generals launched the coup five days ago and overthrew Abdallahi. The coup is also said to be a result of Abdallahi's decision to sack Abdel Aziz and other senior army officers.
Meanwhile, the African Union has decided to suspend Mauritania's membership, the bloc's president said on Saturday.
"The AU will suspend Mauritania's membership until the country returns to constitutional government," said Bernard Membe, the Tanzanian foreign minister.
Tanzania currently holds the AU's rotating presidency.
However, a spokesman for the AU said the country will remain in the group but will not be able to participate in meetings or votes until its "constitutional order" is restored.
Mauritania is no stranger to coups since it gained independence from France in 1960.
Abdallahi's election was hailed as a model of democracy for Africa after a three-year transition following a bloodless coup in August 2005.