She also said the bomber was known "up to a point" by the intelligence services.
Among the first victims to be named were eight-year-old Saffie Roussos from Leyland and teenagers Olivia Campbell, 15, from Bury and Georgina Callander from Chorley. A woman detained on Wednesday was released without charges.
But he said there were still "gaps in our understanding" of the plot, as investigators probed Abedi's potential links to jihadis in Britain, Europe, Libya and the Middle East.
"Even if the families knew about their sons' ideologies or affiliations, they wouldn't talk but keep it to themselves", he said.
British Prime Minister Theresa May has raised the threat level to the highest possible rating, meaning another atrocity is expected imminently. "This is not us".
"I am asking the Department of Justice and other relevant agencies to launch a complete review of this matter, and if appropriate, the culprit should be prosecuted to the fullest extent of the law", he said.
Abedi fled Tripoli in 1993 after Moammar Gadhafi's security authorities issued an arrest warrant and eventually sought political asylum in Britain. He is now a manager of the Central Security force in Tripoli.
"We're now 20,000 police officers down, and we get atrocities like this".
Protected sites in London also include Downing Street, embassies and the Palace of Westminster.
The Chelsea soccer team announced it would cancel Sunday's victory parade in London that was to have celebrated the team's Premier League title win this season.
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He said that while Abedi's family was well-known and respected - and his father even used to perform the call to prayers during the many years he lived in Manchester - Abedi himself was an unsociable young man who didn't regularly attend mosque or community gatherings. Twelve of those rushed to hospital were children.
"There was a policeman, armed policeman, shouting at my neighbor. and I realized there is something wrong here", he said.
"They will be on my mind and in my heart everyday and I will think of them with everything I do for the rest of my life", Grande wrote. "Enormous progress has been made", he said. "I haven't seen police come to his house".
Theresa May is to raise concerns with Donald Trump at a North Atlantic Treaty Organisation meeting later.
Some of the clues suggesting Abedi was part of a wider terror network in Manchester have been revealed in leaked photos of the remnants of the bomb itself.
The Islamic State has claimed responsibility, though British authorities have not determined whether he was truly associated with them or received any assistance while building his bomb or plotting his crime.
British officials said Abedi had been on the radar of the intelligence community before the massacre.
Manchester mayor Andy Burnham said he had complained to the acting American ambassador and had been assured that leaks out of the U.S. would stop.
Her only direct comment since the attack has been a social media posting shortly afterwards in which she said she felt "broken" and "so so sorry". Sixty-four people were injured in the attack.
Officials said all those hospitalized had been identified. Our job is to report the news - not make it by tipping terrorists off that police are hot on their heels.
Sylvia Hui Rob Harris and Angela Charlton contributed.