The ultimatum given to the 93-year-old leader, who has been in power for 37 years, came five days after the military seized power and placed Mugabe under house arrest.
In a rare sign of solidarity between the people and the army, which has often been a pillar of support for Mugabe's rule, thousands of Zimbabweans took to the streets on Saturday to express support for the military's operation and call for Mugabe's removal.
But in his national address, the veteran leader defied the calls to resign, saying he would preside over a ZANU-PF congress next month.
"The party congress is due in a few weeks and I will preside over its processes," said Mugabe.
Al Jazeera's Haru Mutasa, reporting from Zimbabwe's capital, Harare, said "a lot of people are surprised" he did not step down "after all this pressure".
"There is no celebration on the streets, just disbelief," she said.
"Sources behind the scenes say that he is being stubborn and for now he isn’t going anywhere," added Mutasa.
"He could be trying the last few weeks before the congress to go out with a graceful exit. It could be a plan he's working on to exit gracefully, or it could also be that he doesn't understand there is such a massive unhappiness.
"The thing to watch now is Monday midday deadline when, if he doesn’t step down, then they will impeach him in parliament."