* May calls inquiry into disaster as death toll rises
* Opposition leader says truth will come out
* Residents had warned of impending disaster (Adds May calling public inquiry, new quotes throughout)
LONDON, June 15 (Reuters) - British Prime Minister Theresa May on Thursday promised a public inquiry into a fire that gutted a 24-story apartment block killing at least 17 people, as the government faced questions about how such a devastating blaze could have occurred.
Smoke was still wafting out of the blackened shell of the Grenfell Tower on Thursday where specialist search teams faced hazardous conditions as they scoured the wreck, with external cladding still falling from the building.
Fire engulfed the social housing block, where as many as 600 people lived in more than 120 apartments, in the early hours of Wednesday, turning it into a flaming torch in minutes.
"Sadly I can confirm that the number of people that have died is now 17," London police commander Stuart Cundy told reporters.
He said that number was expected to rise and firefighters have said they did not expect to find any more survivors after rescuing 65. Thirty-seven people remained in hospital, with 17 of them in critical care.
"Our absolute priority for all of us is identifying and locating those people who are still missing," Cundy said before declining to comment on speculation about the likely final death toll.
An investigation into the cause of the blaze, the worst in the British capital in a generation, was underway. But the shock at its scale turned to anger and recriminations on Thursday.
Opponents of May's government demanded to know whether more could have been done to prevent the disaster and if spending cuts to local authorities had played a part.
"Right now, people want answers and it's absolutely right and that's why I am today ordering a full public inquiry into this disaster," said May, who visited the scene on Thursday to meet members of the emergency services.
"We need to know what happened, we need to have an explanation of this. People deserve answers; the inquiry will give them."
Local residents say there had been repeated warnings about the safety of the building, which recently underwent an 8.7 million pound ($11.1 million) exterior refurbishment, which included new external cladding and windows.
Planning documents detailing the refurbishment did not refer to a type of fire barrier that building safety experts said should be used when high-rise blocks are being re-clad, according to Reuters research.
The government has also promised that other tower blocks which were also recently refurbished would be assessed.
"We have to get to the bottom of this. The truth has got to come out, and it will," opposition Labour leader Jeremy Corbyn said as he visited volunteers at the site.
London Fire Brigade chief Dany Cotton, who said the fire had been unprecedented in its scale, said urban search units backed by specialist dog teams would slowly make their way through the building as structural surveyors helped make the tower safe.
Survivors, many of whom lost all their belongings in the blaze, spent the night at emergency shelters, as charities and local support groups were flooded with donations of clothes and bedding from shocked Londoners.
Piles of shoes, clothes, duvets and other items accumulated in several spots where volunteers were sorting them. The local authority, the Royal Borough of Kensington and Chelsea, thanked donors but said it could not take any further donations.
Others gathered seeking news of the missing. Semira Mohammed, 37, a science technician who lives nearby, said she knew a family of five who lived on the 21st floor of the block.
"I kept calling and calling," she told Reuters. "The phone was ringing but they didn't reply to us. We were from the same community, and many in the tower were. Maybe 70 percent are from Somalia, Sudan, Morocco, and she was so happy to be in the flat right in that community."
The Syrian Solidarity Campaign said on its Facebook webpage that Mohammed Al Haj Ali, a refugee from Syria, was one of those who had been killed.
Queen Elizabeth said her thoughts and prayers were with those families who had lost loved ones and with the many people still critically ill in hospital. She also paid tribute to the bravery of firefighters who risked their lives to save others.
"It is also heartening to see the incredible generosity of community volunteers rallying to help those affected by this terrible event," the queen said.
Her grandsons, Prince William and Prince Harry, along with William's wife Kate, donated to a relief fund organized by London's local Evening Standard newspaper.
The singer Adele was among many Londoners who gathered close to the burnt tower late on Wednesday to show sympathy for the victims and survivors.
The tower, built in 1974 on the low-rent housing estate in North Kensington, sits in a part of west London surrounded by some of the most affluent areas of the capital.
Accounts emerged of people trapped inside as the blaze destroyed everything around them, shouting for help and trying to escape through windows using makeshift ropes from bed sheets tied together.
By Thursday morning, there was no sign of life in or around the blackened hulk. Security cordons were in place around the base of the tower and the ground was littered with charred debris.
Outside the cordons, impromptu tributes appeared with photos of missing people, messages of condolences, flowers and candles. (Additional reporting by Kate Holton, Michael Holden and Costas Pitas; Writing by Estelle Shirbon and Michael Holden; Editing by Hugh Lawson)