He said the death of Jean Charles de Menezes was a "tragedy", but admitted more people could be shot as police hunt suspected suicide bombers.
The 27-year-old electrician's family condemned the shooting and said there was no reason to suspect him.
Police said on Sunday that the man was arrested in Tulse Hill, south London, on Saturday evening. He is the third man to be questioned under the Terrorism Act.
Home Secretary Charles Clarke described the shooting of Mr Menezes as an "absolute tragedy".
Mr Menezes, who lived in Tulse Hill, was completely unconnected to Thursday's attempted bombings, Scotland Yard has confirmed.
The shooting is being investigated by Scotland Yard's Directorate of Professional Standards, and will be referred to the Independent Police Complaints Commission.
Officers are still searching an address raided on Saturday in Streatham Hill, south London, in connection with the failed attacks.
Sir Ian told Sky News: "This is a tragedy. The Metropolitan Police accepts full responsibility. To the family I can only express my deep regrets."
He said there was no reason to believe the four men sought over the failed bombings - whose images caught on CCTV were released on Friday - had left the country.
He acknowledged "somebody else could be shot" as the hunt continued, but added "everything is done to make it right".
But he said the "shoot to kill" policy for dealing with suspected suicide bombers would remain in force.
JEAN CHARLES DE MENEZES
Born 07/01/78, a Brazilian national
Originally from the town of Gonzaga, 500 miles northeast of Sao Paulo in the south-eastern state of Minas Gerais in Brazil
Moved to Sao Paulo at age 14
Lived in London for three years, working as an electrician
Mr Clarke told the BBC: "I very, very much regret what happened.
"I hope [the family] understand the police were trying to do their very best under very difficult circumstances."
On the ongoing bombings investigation, "good progress" was being made thanks to "tremendous support" from the public.
Mr Clarke said he was postponing joining his family on holiday because of the current crisis.
Mr Menezes's family is struggling to come to terms with his death.
Mr Pereira, from London, told the BBC: "Apologies are not enough. I believe my cousin's death was result of police incompetence."
Describing his cousin as a "person full of life" he said he had been "a victim of government's mistakes".
Mr Menezes was from the city of Gonzaga in Minas Gerais state and had lived in London for over three years.
Mr Menezes' grandmother, Zilda Ambrosia de Figueiredo, told Globo TV "there was no reason to think he was a terrorist".
The body of Mr Menezes is to be taken back to Brazil as soon as possible.
Brazil's foreign minister Celso Amorim met Foreign Office officials in London on Sunday to seek an explanation for the shooting.
'Shocked and perplexed'
"The Brazilian government and the public are shocked and perplexed that a peaceful and innocent person should have been killed," he said.
"Brazil is totally in solidarity with Britain in the fight against terror but people should be cautious to avoid the loss of innocent life."
He spoke by phone to Mr Straw, who he said promised a full investigation into the death.
"I said that was very important. We can't recover the life of the Brazilian citizen who has been killed, but we can discover the details."
He will meet Mr Straw in person on Monday evening.
The BBC's correspondent in Brazil, Tom Gibb, said Mr Menezes had lived for a time in a slum district of Sao Paulo and that could explain why he had run from the police.
1: Jean Charles de Menezes leaves a house under surveillance and arrives at Stockwell station
2: Witnesses say he vaults the automatic ticket barriers and heads for the platforms
3: He then ran down an escalator after being approached by up to 20 plain-clothed police officers and tried to board a train
4: He apparently refuses to obey police instructions and after running onto a northbound Northern line train, he is shot dead