The killing of General Athanase Kararuza and his wife were "shameful acts," he said.
Their daughter was also wounded in the attack outside her school in the capital, Bujumbura.
More than 400 people have been killed in unrest since Mr Nkurunziza announced a year ago his bid for a third term.
He won elections in July, after surviving a coup attempt and mass protests to demand that he steps down.
General Kararuza was the latest in a string of high-ranking army officials who have been targeted by unidentified gunmen.
"He energetically fought against the coup plotters last year and exceptionally contributed in strengthening peace and security during and after elections," Mr Nkurunziza said in a statement.
General Kararuza's bodyguard was also killed in the shooting.
Security forces have also been accused of killing opponents and dumping them in mass graves by rights group Amnesty International.
Shortly after the attack in Bujumbura, the International Criminal Court announced that it was starting a preliminary investigation into the violence in Burundi.
This will decide whether a full-blown investigation will take place, which could result in charges against those accused of being behind the violence.
BBC Great Lakes reporter Robert Misigaro says the killing is a blow for President Nkurunziza because it shows that he cannot guarantee the safety of his officers.
On Sunday, the Minister for Human Rights Martin Nivyabandi survived a grenade attack as he was coming out of church.
Although both opposition and government forces are ethnically mixed, some fear that the violence could descend into a repeat of the genocidal killings which the country has previously experienced.
Mr Nkurunziza is the former leader of a Hutu rebel group which battled a Tutsi-dominated army for many years until he came to power in 2005 as part of a peace deal.
The African Union had said it would send a 5,000-strong peacekeeping force to the country even if the government did not accept it but it has since back-tracked.

The lorry was stopped just outside Armagh on Wednesday night.
A police officer told Newry Court that Conor Toal, 30, from Oakridge in Armagh city, was spotted passing objects from his car to the lorry.
It was driven by Darren Donnelly, 32, from Willowmount Avenue in Omagh, County Tyrone.
A subsequent search of the lorry led to the discovery of the money.
The officer said he believed the cash was being transported to England, where it was to be used by a major crime gang to buy illegal drugs to bring back to Northern Ireland.
A later search of Mr Toal's home uncovered another £4,000 in cash and four luxury watches, including two made by Rolex.
Both men denied charges of possessing criminal property, transporting criminal property and entering into an arrangement.
The judge refused bail because he said he feared the pair could interfere with the police investigation, which is at an early stage.
They are due to appear again via videolink at Armagh Court on 13 September.
Burundi's President Pierre Nkurunziza has given the security forces a one-week deadline to find the killers of a senior general shot dead on Monday.
[NEXT_CONCEPT]
Two men have been remanded in custody after being charged following the discovery of more than £400,000 in cash inside a lorry.