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.
Mrs May did not reveal how all the money would be raised, but said Chancellor Philip Hammond would set out the details "in due course".
Theresa May's claim of a "Brexit dividend" for the NHS is a "reckless pledge" that ignores the economic damage leaving the European Union could wreak, a group of leading health professionals say today.
British Prime Minister Theresa May has announced a significant funding increase for the country's beleaguered National Health Service.
Scottish Secretary David Mundell said: "In the NHS's 70th year, I urge the Scottish government to invest this extra money in improving health services in Scotland". Seventy years on, it remains this country's most valued public service, an institution that is there for every family, everywhere, at the best of times and at the worst.
"That money will be coming back".
May said the extra funding will help "secure" the NHS's future.
That could include reducing spending elsewhere.
But the PM is under continuing pressure to explain where the money will come from to pay for the rise.
Despite leaving, Britain will continue to make payments to the European Union over several decades to settle an exit bill of around 39 billion pounds.
"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". The specifics of how this will work haven't yet been outlined.
The party called for "every penny" of the additional funds to be spent on the NHS. What kind of tax increase this would amount to remains a Cabinet secret. Tax rises? Borrowing? What's plan B?
Will there be enough to deliver the progress demanded by the PM on cancer outcomes, mental health treatment and waiting time targets routinely missed?
Mr Davis questioned the Scottish politician on the viability of Mrs May's cash injection following years of austerity implemented in the wake of the finance crisis of 2008.
As health professionals experiencing the effect of chronic under-investment in our health service we welcome the Prime Minister's pledge of greater investment for the NHS.
'However, we will be scrutinising the detail of this new package of funding to assess the degree to which it will make a real difference in both the short and long term to frontline patient care'.
But she remained vague on how the new money, totalling £394m more a week in real terms by the end of the settlement, will be paid for, beyond mentioning taxes and repeating her confidence in a so-called Brexit dividend.
"This additional United Kingdom government investment in Scotland has the potential to make a real difference for people in Scotland".
The IFS has estimated that there will be a £3 billion gap in social care spending by 2023/24.
"The important point is that you can only afford to fund the NHS well if you have a strong vibrant and dynamic economy where the government is focused on enterprise and growth, so that's why I think we're able to do it", he told reporters.