Cllr Feilding-Mellen, who was forced to flee his £1.2m home after allegedly receiving threats from residents outside, said in a statement: "I have always tried my utmost to meet my responsibilities with integrity, hard work, and a commitment to serving the interests of all residents of the borough.
"It has been suggested several times since the tragic event of the 14 June that I should resign, but until now I have felt that it was my duty to do whatever I could personally to back the council's efforts to help the fire's victims, to encourage all of the dedicated officers within the council as they worked tirelessly for the relief effort, and to support the council's leader."
Several media organisations obtained an emergency court order demanding the meeting remain open to the public and journalists when the council tried to make it private.
They claimed they had to shut it down because speaking about the fire in public might "prejudice" the inquiry in how the fire was started which has been set up Theresa May.
It was the first public meeting of the council's cabinet since the fire earlier this month.
In Pictures: Grenfell Tower after the fire
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Responsible for the relief effort has been passed to a coalition of several neighbouring councils, the Metropolitan Police, the London fire brigade and the Red Cross after the council was condemned for its "chaotic" response.
Residents have told of receiving overwhelming and help and support from volunteers who flocked to a local community centre but say they had almost complete silence from the council in the first few days after the fire.
Opposition councillors said the council had repeatedly failed to give out accurate information to residents looking for loved ones or a place to stay and that there was a lack of staff on the ground providing advice.
The council have also been heavily criticised for its decision to use a cheaper, more flammable version of the cladding placed on the outside of the block during a 2016 refurbishment.
The aluminium cladding is believed to have spread the fire after a fridge explored on the fourth floor. A non-flammable zinc-based version was also discussed but rejected after it would cost £293,368 more, according to 2014 documents.
The memos, seen by The Times, sent between project management consultants Artelia UK show them apparently coming under political pressure to find a way to reduce costs with little mention of fire safety concerns.
An "urgent nudge email" about the price of cladding sent by the managers of Grenfell Tower, Kensington and Chelsea Tenants Management Organisation, said they needed "good costs" to present to Cllr Feilding-Mellen.
Cllr Feilding-Mellen, a former land developer, was chair of the housing committee and oversaw the refurbishment project.
Additional reporting by PA
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