The west London airport this week introduced a daily 100,000 passenger limit, meaning flights must be rescheduled to other airports or cancelled.
On Tuesday, it appealed to airlines to stop selling summer tickets as it imposed the cap, which will remain until September 11.
Chaotic scenes at terminals have led to long queues, with passengers lining up outside during a heatwave, mountains of luggage not returned to customers and thousands of flights scrapped. On Wednesday, a US Delta plane carrying only luggage flew out of Heathrow after it was forced to improvise to reunite passengers with their possessions.
In a statement, the airline accused Heathrow bosses of showing "blatant disregard for consumers" by attempting to force it to "deny seats to tens of thousands of travellers".
It said Heathrow's demands were "unreasonable and unacceptable" and described the airport management as "cavalier about travellers and airline customers".
While Emirates had planned ahead for the reopening of the skies, Heathrow "chose not to act, not to plan, not to invest. Now faced with an 'airmageddon' situation due to their incompetence and non-action, they are pushing the entire burden ― of costs and the scramble to sort the mess ― to airlines and travellers".
Emirates, which operates six daily return flights between the airport and Dubai, continued: "LHR [London Heathrow] last evening gave us 36 hours to comply with capacity cuts, of a figure that appears to be plucked from thin air.
"Their communications not only dictated the specific flights on which we should throw out paying passengers, but also threatened legal action for non-compliance.
"This is entirely unreasonable and unacceptable and we reject these demands."
Emirates said it would continue business as usual to and from Heathrow, "until further notice".
It said its ground-handling and catering crews were ready and capable of managing planned flights, and moving to alternative UK airports at short notice was not realistic.
"Ensuring ground readiness to handle and turn around a widebody long-haul aircraft with 500 passengers on board is not as simple as finding a parking spot at a mall," it said.
"From our past 10 months of regularly high seat loads, our operational requirements cannot be a surprise to the airport."
It suggested shareholders scrutinise the decisions of the airport's management team.
The airline said 70 per cent of its passengers from Heathrow were booked to travel on connecting flights from Dubai, highlighting the effect the rejected cuts would have on its own operations. It also pointed out the order was being made "during the super peak period with the upcoming UK holidays, and at a time when many people are desperate to travel after two years of pandemic restrictions".
Virgin Atlantic also criticised the airport’s actions and claimed it was responsible for failures which are contributing to the chaos.
British Airways announced it will cancel six additional daily short-haul flights over the next fortnight in response to the cap.
It has already axed tens of thousands of flights this summer.
"As a result of capacity limits imposed by Heathrow Airport, Etihad Airways has been forced to reschedule flight EY26 from London to Abu Dhabi on Thursday, July 14," an Etihad spokesman said.
Many of the problems at Heathrow and other European airports are being caused by staff shortages, from security and check-in staff to ground crew, after the industry struggled to recruit sufficient numbers once Covid travel restrictions were removed.
Heathrow's decision to limit numbers to 100,000 passengers a day means more than 4,000 seats will be scrapped every day. This comes on top of tens of thousands of passengers whose flights have already been cancelled. On Monday alone, Heathrow ordered airlines to ditch 61 flights, affecting about 10,000 passengers.
About 131,000 passengers departed the airport on August 4, 2019, Heathrow’s busiest day on record.
Responding to Emirates’ refusal to cancel flights, a Heathrow spokeswoman said: “Aviation is a complex network and no one can operate in isolation. The network continues to suffer from Covid-related challenges.
“While many factors have resulted in the delayed flights, misconnected bags, long waits for arriving bags and last-minute cancellations at Heathrow and airports across Europe in recent weeks, a key issue is airline ground-handling teams, which are currently only resourced up to 70 per cent capacity to serve passenger demand that has returned to 80 per cent to 85 per cent of pre-pandemic levels.
“For months we have asked airlines to help come up with a plan to solve their resourcing challenges, but no clear plans were forthcoming and with each passing day the problem got worse.
“We had no choice but to take the difficult decision to impose a capacity cap designed to give passengers a better, more reliable journey and to keep everyone working at the airport safe.
“We have tried to be as supportive as possible to airlines and our 100,000 cap on daily departing passengers is significantly higher than the 64,000 cap at Schiphol [in Amsterdam].
“It would be disappointing if instead of working together, any airline would want to put profit ahead of a safe and reliable passenger journey.”
Virgin Atlantic chief customer and operating officer Corneel Koster said “everybody should have got ready for this increased demand”.
He told the PA news agency: “If you’re around the table and the Heathrow voice says ‘it won’t happen, it will come later, I will only open my fourth terminal in June’, that’s a planning mistake.
“They have downplayed demand. They should have opened T4 earlier.
“They should have played an even stronger role in the community.”
In an open letter to passengers on the capacity cap this week, Heathrow chief executive John Holland-Kaye, who has previously said problems could last 18 months, apologised again and admitted scenes in terminals had been unacceptable.
He said the government move to suspend its "use it or lose it" rules on flight routes had helped by allowing flights to be cancelled in advance, but it had not been enough — some airlines were still scheduling too many flights, he said. The policy had led to ghost flights because most airlines would rather fly empty than lose lucrative routes.
The government has ordered airports to review their schedules to give passengers confidence that their travel plans would not be disrupted.
In December last year, Heathrow said it expected passenger numbers for 2022 to reach around 45 million.
It subsequently raised its forecast to “nearly 53 million” in May, and 54.4 million in June.
Terminal 4 was only reopened on June 14, some three months after the UK lifted all remaining coronavirus travel restrictions.
It was the last terminal at a major European airport to resume operations during the pandemic.