Al-Bashir, who signed a peace treaty with Garang in January, announced the formation of a national committee in cooperation with the Sudan People's Liberation Movement (SPLM) to investigate the crash.
He regretted the death of Garang and the violence that has claimed the lives of dozens of Sudanese so far.
"We offer our condolences to the people of Sudan and reconfirm our determination to go on with the peace process" he said in a speech aired live by Aljazeera on Wednesday.
He said some parties were trying to hinder the peace process and appealed for calm. However, he did not name any of the parties.
Violence in Khartoum erupted on Monday when angry southerners took to the streets after the death Garang, who fought the northern government for two decades before making peace.
"There are quite a number of casualties and it's quite serious," UN spokeswoman Radhia Achouri said.
Some northerners responded to Monday's looting and attacks by southerners by forming vigilante groups, some carrying guns. Those groups have been roaming the streets despite a curfew.
Salva Kiir, who has swiftly been installed as the new head of the SPLM, echoed Bashir's call for calm in the southern settlement of New Site where he met top US and South African envoys on a diplomatic push to maintain the fragile peace pact.
Kiir [L] echoed al-Bashir's
call for calm
"Enemies of peace may want to take opportunity of this situation," Kiir said. "We are appealing to all the Sudanese people to refrain from any hostility."
Police put the death toll at 46 on Tuesday. The United Nations, in a security briefing on Wednesday, said 20 people had been killed overnight.
The International Committee of the Red Cross (ICRC) said it counted 84 bodies in a morgue, all killed since Monday, but it was not immediately clear if they included those killed in the most recent clashes on Wednesday.
The violence has raised fears that fresh north-south tensions could undermine a January peace deal between Garang's former rebel movement and the northern government.
"Peace is being jeopardised in the short run," the top UN envoy in Sudan, Jan Pronk, said. "Those people who were pleased with the peace -- by not behaving -- are endangering the very thing they were pleased with."
Southerners fear the absence of Garang, who became first vice president on July 9 under the peace agreement, could weaken their hand in governing the oil-exporting nation.
Sudan is divided between an Arabised Muslim north and the south that is a mix of ethnicities who embrace Christianity, animism and Islam.
Wednesday's violence, that began in the capital's suburbs, spread to a downtown area following rumours that a southern militia leader had been killed. The leader later appeared on television to refute the talk.
Streets were full of cars heading out of the city centre, and five truckloads of soldiers and riot police headed into the central residential and commercial area, witnesses said. Gunshots were also heard and teargas was fired, witnesses said.
"Enemies of peace may want to take opportunity of this situation, we are appealing to all the Sudanese people to refrain from any hostility."
Salva Kiir, new SPLM chief
Garang's funeral will be on Saturday in Juba in the south, after his body tours other towns in the area. On Tuesday, grieving relatives and supporters paid respects to Garang around a simple bed.
The conflict in southern Sudan began in 1983 when the Khartoum government tried to enforce Sharia Islamic law on the region. Two million people were killed, mainly by hunger and disease.
The peace deal included giving southerners the right to vote on secession after a six-year interim period and shared out Sudan's oil wealth between north and south roughly equally.
Sudan also faces continued civil strife in its western Darfur region, which has killed tens of thousands and forced out around 2 million from their homes.