The British Medical Association and other unions said their staff "have literally kept the country alive for the past year".
In an open letter to the chancellor they called for a "fair pay deal".
Health Secretary Matt Hancock earlier defended the pay increase, saying it was "what we think is affordable".
News of the pay deal emerged earlier this week, when the Department of Health and Social Care officially recommended the 1% pay rise to the independent panel that advises the government on NHS salaries.
The panel is due to make its own pay recommendations in early May, when ministers will make their final decision.
- How much will NHS staff be paid?
But healthcare staff reacted with anger at the plan. The Royal College of Nursing called the rise "pitiful" and started preparing for strike action, saying that its members should get 12.5% instead.
And Unite - the third largest union in the NHS - said it is considering a strike ballot.
Now more unions have joined their calls. The BMA, the Royal College of Nursing, the Royal College of Midwives and Unison have written a joint letter to Chancellor Rishi Sunak to express their "dismay".
"The proposal of a 1% pay offer, not announced from the despatch box but smuggled out quietly in the days afterwards, fails the test of both honesty and fails to provide staff who have been on the very frontline of the pandemic the fair pay deal they need," the letter said.
"Our members are the doctors, nurses, midwives, porters, healthcare assistants and more, all ready exhausted and distressed, who are also expected to go on caring for the millions of patients on waiting lists, coping with a huge backlog of treatment as well as caring for those with Covid-19."
- The lowest minimum full-time salary - for newly employed drivers, housekeeping assistants, nursery assistants and domestic support workers - is £18,005 per year
- The starting salary for most newly qualified nurses is £24,907
- Staff in "high-cost areas", such as London, get extra payments
"It is very disappointing that the government has said that a 1% pay rise is all that is affordable when they know that the assumption was that the 2021/22 NHS pay rise would be 2.1% - and that this was covered by the NHS revenue settlement announced by Theresa May in June 2018," said Saffron Cordery, the deputy head of NHS Providers.
Although these assumptions were published in 2019 before the pandemic, NHS Providers said the last year has "strengthened the case for a larger pay rise for NHS staff".
The government has repeatedly insisted that a 1% increase is all they can afford at a time when public finances are already stretched due to the pandemic.
Speaking at Friday's Downing Street briefing, Mr. Hancock said the pay award reflected the "difficult financial circumstances the country is in".
"One of the challenges we've faced as a country is in terms of the financial consequences of the pandemic," he said.
"We've proposed what we think is affordable to make sure in the NHS people do get a pay rise."
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The pay award would cover nearly all hospital staff, but not GPs and dentists.
In addition, some staff whose pay band is being changed will get more than a 1% increase as a result of a previously agreed three-year pay deal.
The government said newly-qualified nurses got a 12% pay rise over the course of that three-year agreement They added that 1% was a "real-terms increase", as the latest official inflation figure was 0.9%.
But Unite's national officer for health said it would turn into a "pay cut in real terms" if inflation rises over this year.
The boss of the Royal College of Nursing, Dame Donna Kinnair, also said the rise shows ministers are "dangerously out of touch with nursing staff, NHS workers and the public".
Meanwhile, Labour has said NHS "heroes" deserved more money and the proposed increase was "nothing short of an insult".
media captionThe Labour leader says NHS staff and other public sector workers "need to be properly rewarded"
The NHS in Scotland, Wales, and Northern Ireland is run by devolved administrations.
The Scottish government has announced that 2021-22 pay negotiations will be delayed until the summer because of the disruption caused by Covid. The staff has been given an "interim" pay rise of 1%, which will form part of the new settlement.
NHS workers in Northern Ireland were promised a one-off £500 "special recognition" payment in January, following a similar announcement in Scotland.
The Welsh government has said it will not set a "ceiling" of 1% on NHS pay rises for 2021-22.