In response, Downing Street said it "greatly valued" ambulance workers and the "door remains open" for talks.
Ministers have accused workers of putting lives at risk during strikes.
GMB represents more than 10,000 ambulance workers who went on strike on Wednesday across nine ambulance services in England and Wales.
Steve Rice, writing on behalf of the GMB Ambulance Committee, said he was "appalled" at recent government statements and accused ministers of attempting to remove workers' right to strike .
He wrote: "We feel utterly betrayed by the way your government has singled out ambulance workers as part of a crude attempt to remove our right to strike.
"You and your ministers should be ashamed of the way you have tried to paint us as uncaring about safety standards - nothing could be further from the truth.
"We want a constructive relationship with government - to talk about pay and seriously improve conditions throughout the ambulance service. But you are making us and our ambulance colleagues feel demonised."
Mr Rice concludes: "Please talk to us and stop attacking us."
The letter comes after the prime minister described the industrial action as "terrifying".
He told the Commons on Wednesday: "What is terrifying is that right now people do not know whether, when they call 999, they will get the treatment they need."
Business Secretary Grant Shapps told MPs that "a lack of timely co-operation" from unions was "putting their constituents' lives at risk."
Responding to the GMB letter, Downing Street said the health secretary's door remains open to talks with unions and it has accepted the independent pay review body's recommendation to give one million NHS workers a pay rise of at least £1,400 this year.
A spokesperson said: "However, we must keep the public safe, which is why we are introducing minimum service and safety levels across a range of sectors to ensure that lives and livelihoods are not lost."
Mr Sunak said he hoped to "find a way through" the deadlock with unions to avert further industrial action and touted the minimum service level legislation.
The Bill is due to considered by MPs again on Monday.
Labour deputy leader Angela Rayner tweeted: "Ministers know this shoddy, unworkable Bill won't do a thing to help working people or avoid strikes."