Speaking to reporters in Istanbul, Ala identified the attacker as Mehmet Ozturk, who was born in 1992 and registered in Gaziantep, a city in southern Turkey, near the Syrian border.
"The evidence at hand shows that he is connected to the Daesh terrorist organization," said Ala, using another name for ISIS.
"We are still considering with great sensitivity other connections or forces behind this. The fight against terror will continue."
He said that security forces had said there had been no previous warrants issued for the bomber. Five others had been detained in relation to the attack, he said.
Two of the four dead were American-Israeli dual citizens, according to an Israeli government source and a U.S. source.
A third Israeli was also killed in the attack, Israel's Foreign Ministry said, and Iran state media reported that an Iranian was killed.
In addition to the fatalities, 36 people were injured in the blast, Turkish Health Minister Mehmet Muezzinoglu said.
"I condemn those who are killing humanity like this," Muezzinoglu said.
Among the injured were 11 Israelis, officials said. It was unclear whether the suicide bombing was targeting Israelis, Prime Minister Benjamin Netanyahu said.
Since August 2014, Israel has advised its citizens to avoid nonessential travel to Turkey.
U.S. National Security Council spokesman Ned Price said the United States is steadfast in its support for Turkey.
"These repeated acts of terrorism in Turkey must come to an end," Price said in a statement
. "We are in close touch with Turkish authorities and reaffirm our commitment to work together with Turkey to confront the evil of terrorism."
Also on Sunday, a soccer match in Istanbul between two Turkish soccer teams, Galatasaray and Fenerbahce, was postponed upon agreement from both teams, according to a statement from the Governorship of Istanbul. Authorities cited "serious intelligence" as the cause but didn't specifically link their decision to the explosion.
No claim of responsibility
Police cordoned off Taksim Square after the attack, as helicopters flew overhead and multiple ambulances gathered at the site.
Stunned shoppers ran away from the scene, some in tears.
No group immediately claimed responsibility.
NATO Secretary General Jens Stoltenberg condemned the attack as "yet another terrorist outrage targeting innocent civilians and our ally Turkey."
"There can be no justification for terrorism," he said in a statement. "NATO allies stand with Turkey, united in our determination to fight terrorism in all its forms."
Days after deadly car bomb in Ankara
The attack came nearly a week after a car bomb ripped through a busy square
in the capital, Ankara, on March 13.
The Kurdistan Freedom Falcons, or TAK -- a militant offshoot of the Kurdish separatist group, PKK -- boasted that it was behind that bombing, which killed 37 people. The group said on its website Thursday that its militants struck "in the heart of (the) fascist Turkish republic."
The PKK, or Kurdistan Worker's Party, seeks an independent state in Turkey and has been in an armed struggle with the government for decades. The United States and EU have designated both it and the TAK
as terrorist organizations.
A ceasefire between the PKK and Turkey fell apart last summer, after which Turkey bombed the terror group's positions in northern Iraq while also imposing curfews in crackdowns on heavily Kurdish areas in southeastern Turkey.
Some residents have accused Turkey of unjust collective punishment, saying security forces have acted with impunity and killed civilians.
Kurdish militant group warned tourists
The Kurdistan Freedom Falcons specifically have been tied to a number of horrific attacks on their own.
This includes a February bombing targeting military vehicles
in central Ankara that killed 28 people.
The TAK called that attack "revenge" for Turkish military actions and threatened more violence -- warning foreigners, especially, to stay away from Turkey.
"Tourism is one of the important sources feeding the dirty and special war, so it is a major target we aim to destroy," the TAK said at the time.