At least 71 people were injured in the attack on the popular market.
Duterte described the attack as an act of terrorism, and declared the nation in "a state of lawlessness," the official Philippines News Agency reported, authorizing the police and military to search cars and frisk people at checkpoints.
He said he had not declared martial law, according to PNA.
"We have to confront the ugly head of terrorism," Duterte said Friday, standing near the explosion site in his hometown. "We will take this as a police matter about terrorism."
The cause of the explosion, which happened around 10 p.m. ET Friday, is not known. But presidential spokesman Martin Andanar said components of a suspected improvised explosive device were found at the scene, according to CNN affiliate ABS-CBN.
No group has claimed responsibility, but Duterte said it's possible the explosion "could be a reprisal" from extremists.
Philippine's National Defense Secretary Delfin N. Lorenzana said he "assumes" the attack was carried out by the Islamist militant group Abu Sayyaf.
Duterte, the longtime mayor of Davao City, was elected President in May. He campaigned on a no-nonsense approach to crime and launched an intense -- and deadly -- crackdown on drug dealers.
The Philippine Daily Inquirer's "Kill List" -- regarded as one of the most accurate records of the killings of suspected drug dealers by police and vigilantes -- has recorded 832 deaths since Duterte assumed office June 30. Police say at least 239 drug suspects were killed in the three weeks after Duterte's inauguration.
The government's heavy-handed tactics
have drawn international criticism. Many public officials have been accused of being involved in the drug business.
And government troops have been battling Abu Sayyaf, which remains outside the country's sputtering peace process.
The group aims to establish an independent Islamic state
on the southern island of Mindanao, where Davao City is located.
Abu Sayyaf is a violent extremist group that split from established Philippines separatist movement Moro National Liberation Front in 1991. It was formed by Abdurajak Abubakar Janjalani, who trained in the Middle East and reportedly met with al Qaeda leader Osama bin Laden.
'This is not a fascist state'
Duterte's spokesman said the constitution gives the President the power to "call out ... armed forces to prevent or suppress lawless violence, invasion or rebellion."
The President can only impose martial law in case of invasion or rebellion, the statement said.
Duterte said people should submit to searches and frisking at checkpoints for the sake of public safety.
"We know that this is not a fascist state. I cannot control the movement of the citizens of the city and every Filipino has the right to enter and leave Davao. It is unfortunate we cannot stop and frisk anybody for just any reason," he said.
Sara Duterte, the President's daughter and the mayor of Davao City, issued a statement urging all citizens to report suspicious activity to the authorities. She also sent a message to other officials.
"I would like to remind other officials to stay within the bounds of their official duties according to their position," she said. "Please leave me in peace to do my job and I'll leave you to focus on your own work."
'I am really scared'
Leonor Rala, a 19-year-old medical technology student at San Pedro College, told CNN she was in her dorm and about to go to bed when she heard an explosion.
She said she initially thought something had fallen on the roof of a neighboring building. She went down to survey the scene of the blast, about 100 yards from her dorm. Emergency teams were already in place.
"I am really scared to go out," she said. "The roads are closed and nobody's allowed to go out of the city. There are bomb threats everywhere and some of my schoolmates are victims of the explosion and now dead."
She continued: "We're very terrified because Davao City was known to be the safest city in the Philippines and a situation like this is very rare."
Witness Janoz Laquihon told CNN Philippines he was at the scene when the explosion happened.
"I saw some smoke. I thought it's just barbecue. A few minutes later ... a big blast."
Witness Father Jboy Gonzales told CNN Philippines that he saw more than 30 people being loaded onto ambulances.
"[A] lot of people are wounded, shocked, traumatized," he said.
Duterte made his name in politics as the mayor of Davao City. His term in office was noted for his hardline stance on drug crime that he has now incorporated into his national policies
. It has resulted in more than 1,900 people being killed in a crackdown that has drawn criticism at home and abroad.
Maria Ressa, executive editor of Philippines news website Rappler
, said the blast occurred amid tensions surrounding Duterte's war on drugs as well as an ongoing peace process with Muslim militants in the southern Philippines.