"Our basic mission is to maintain the country's stability and security," the Lieutenant-General told a news conference.
"We will not allow any breach of security anywhere."
Protesters angry that army commanders have taken control after removing the veteran leader defied a night-time curfew to keep up four months of mass demonstrations on Friday.
Protest leaders dismissed the transitional military council formed by top brass after they toppled Mr Bashir on Thursday, as the "same old faces" from the old regime which had ruled the country with an iron fist for three decades.
Demonstrators demanded a civilian body to lead the transition to democracy and bring an end to the multiple conflicts which have pushed the country into worsening poverty.
Most shops and offices were closed on Friday which is the day of prayer and rest in Sudan.
Vast crowds were expected to throng the streets of Khartoum and its twin city Omdurman after the main weekly Muslim prayers at noon raising fears of confrontation.
Washington called on the military council "to exercise restraint and to allow space for civilian participation within the government".
The European Union urged the army to carry out a "swift" handover to civilian rule.
Sudan's last elected prime minister, opposition Umma party leader, Sadiq al-Mahdi, who was overthrown by Mr Bashir in a military coup in 1989, was expected to address supporters after prayers at one of Omdurman's most revered mosques.
Since returning to Khartoum from self-imposed exile, Mr Mahdi has allied his party with the grass-roots who were the driving force behind the mass protests that preceded Thursday's coup.
The Sudan Professionals Association -- a well organised umbella group of doctors, teachers and engineers -- has called for a huge turnout at Friday's turnout as has the wider opposition Alliance for Freedom and Democracy.
Despite warnings from the new military council to respect the nightime curfew, the soldiers posted outside army headquarters made no move to disperse the protesters who camped out there for a sixth straight night on Thursday, demonstrators said.
Protesters were seen chatting with soldiers. They said their quarrel was with the commanders who had led the coup, not the rank and file.
"There was no difference between last night and previous days and nights for us," said one protester who gave his name as Abu Obeida.
"This is now our square. We have taken it and won't leave until victory is achieved. We broke the curfew. We will continue doing it until we have a civilian transitional government."
The opposition Sudanese Congress Party called for the establishment of a joint military and civilian council to rule for a four-year transitional period.
It also urged the army to give executive powers to civilians.