She said her visits to London and Manchester -- the latter the site of a terrorist attack last month - have highlighted the people who offer comfort and support to those affected.
The prime minister has committed £5m for clothes, food and emergency supplies but was heckled with chants of "coward" after she met survivors of the fire.
Her Majesty was shown the food supplies donated to those made homeless by the fire when she visited the West Way sports centre.
This year, however, it is hard to escape a very' sombre national mood.
Meanwhile, May, stung by criticism, was scheduled to meet victims, volunteers and community leaders in Downing Street on Saturday afternoon, even as angry protests continued at the site and elsewhere in London.More news: SSP Group PLC (SSPG) Insider Buys £128.79 in Stock
The monarch released a statement as she celebrates her official birthday with the Trooping the Colour on Saturday June 17. "United in our sadness, we are equally determined, without fear or favour, to support all those rebuilding so awful affected by injury and loss".
May was criticised for not speaking to Grenfell Tower residents on her first visit following the blaze.
Saturday's newspapers conveyed the sense of anger.
"What I'm now absolutely focused on is ensuring that we get that support on the ground", May said in a BBC Newsnight interview.More news: FIFA Confederation Cup: All eyes on Russian Federation
The Daily Telegraph claimed left-wing militants had been hijacking the demonstrations and spreading political fake news conspiracy theories about the blaze.
People watch from a balcony as people walk during a demonstration following the fire at Grenfell Towers that engulfed the 24-storey building on Wednesday morning, in London, Friday June 16, 2017. "As prime minister, I will be responsible for implementing its findings". "We do not want another fatality arising out of this tragedy", he said.
Seven Moroccans are among the dead, the north African country's foreign affairs ministry said. Relatives of those missing after a high-rise tower blaze in London are searching frantically for their loved ones, as the police commander in charge of the investigation says he hopes the death toll will not rise to three figures.
Pope Francis sent a message of "heartfelt condolences" in which he invoked God's blessings of "strength and peace" on the local community.
In the fire's aftermath, the London Underground on Saturday suspended subway service between Edgware Road and Hammersmith on two lines at the request of the London Fire Brigade.More news: Andy Murray 'wants to keep going' after reaching French Open semi-finals
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