Amin Abdullah, 41, died after dousing himself in petrol outside the London palace days before his appeal hearing.
He had been in a suicidal state after losing his job and was admitted to hospital for treatment.
Imperial College Healthcare NHS Trust said it will "engage fully with any investigations" into the death.
Mr Abdullah was dismissed from Charing Cross Hospital in December for gross misconduct after writing a letter in support of a colleague after a patient complaint.
It is understood he was one of 17 people to sign a petition.
Terry Skitmore, Mr Abdullah's partner, said the letter was mistakenly attached to the back of the petition.
The nurse was charged with writing an "untrue letter" and not using the correct complaints procedure.
After slipping into a severe depression, Mr Abdullah was admitted to St Charles Hospital in Ladbroke Grove for treatment, but had been allowed out to collect a change of clothes.
He was later found on fire by police outside the London home of the Duke and Duchess of Cambridge on 9 February.
Mr Skitmore said Mr Abdullah had an "exemplary background" - winning an award for excellence when he graduated from Buckinghamshire New University.
"I want to get the message out there about the way that the whole thing was terribly handled - it was badly handled because of a bad culture that they have. It just seems to be that if anyone steps out of line they are literally just crushed into the ground," he said.
"They treated him disgustingly - it pushed him into the depression and got us in the position we are in today."
Imperial College Healthcare NHS Trust said it was "saddened to hear of Mr Abdullah's death" and would "engage fully with any investigations".
Central and North West London NHS Foundation Trust, which runs St Charles Hospital, said it would "get to the bottom of what happened" and take any necessary action.
A nurse who set himself on fire outside Kensington Palace after being sacked was "treated disgustingly", his partner has said.