Mrs May also signalled she could tear up some of the changes to NHS structures put in place by former health secretary Lord Lansley as part of an effort to reduce red tape.
"The Tory claim that the NHS can be funded by a Brexit dividend is simply not credible as the United Kingdom will be paying £40 billion to leave the EU", Ms Robison told the BBC's Good Morning Scotland.
The announcement for England means the rest of the United Kingdom will also be given extra money, although it is up to the governments in Wales and Scotland to decide exactly how that is spent.
The newspapers said the 384-million-a-week pledge was politically significant from May - who campaigned against Brexit in 2016 and has been under pressure from hardline Brexiteers ever since to prove her conversion to the cause - because it went above and beyond 350 million.
In response to the announcement of £20 billion for the NHS over the next five years the Chair of the Health Select Committee, Dr Sarah Wollaston MP, said that while she welcomed the announcement it would "not deliver as planned without attention to and uplifts for public health (prevention), social care, workforce training & capital/transformation budgets".
"The economic damage of Brexit could cut up to £3.7bn a year from public services in Scotland by 2030 compared with staying in the EU".
As we prepare to celebrate the 70th birthday of the NHS, the truth is that Brexit poses the biggest risk to our health service in its history.
Speaking at the Royal Free Hospital in north London on June 18, May announced that British taxpayers would need to contribute "a bit more" but promised contributions would be calculated in a fair and balanced way.
"We are clear that there will be an increased burden of taxation", he confirmed. Two thirds of voters said they would be willing to pay one penny more in the pound in income tax to help the NHS, according to a YouGov survey.More news: OPPO Find X with All Screen Design, Pop-up Cameras Launched
He said the financial settlement with the EU, plus the UK's commitments to replace EU funding, "already uses up all of our EU contributions" for the next few years.
In meetings with Philip Hammond, he stressed the chronic needs of the NHS and the importance of every tenth of a percentage point, before withdrawing to let Mr Hunt and the chancellor thrash out the detail.
"These will still be tough years ahead", he added.
"It is too bureaucratic, inhibits joined-up care, and takes money and people away from the front line", she said.
"In five years' time, we don't know exactly what will be happening".
An official spokeswoman from May's Number 10 Downing Street said she did not have the details available.
Johnson, who was visiting Geneva to address the U.N. Human Rights Council, dismissed the idea that many people thought relying on a "Brexit dividend" was nonsense. "We want to see this long-term plan for the NHS delivering for the NHS".
Meanwhile, councils have questioned why the funding announcement did not also include more money for social care and public health, which covers everything from stop smoking services to obesity prevention.