In a tweet criticising US Democrats pushing for a universal health system, he said "thousands of people are marching" in the UK because the NHS is "going broke and not working".
This was believed to be a reference to a Save the NHS march on Downing Street on Saturday, demanding more funding.
Downing Street said Theresa May was "proud" of the UK's system.
Her spokesman said the NHS had recently been ranked as the world's best healthcare system and that Health Secretary Jeremy Hunt, who has hit back at the US president on Twitter, "speaks for the government".
Organisers of the demonstration said it had been aimed at showing people's "love" for the NHS.
The chief executive of NHS England said the US president had "got the wrong end of the stick".
In the UK, the NHS is funded out of general taxation, so people do not have to pay when they get treatment.
In the US, when people get treatment they have to pay, most often through health insurance providers.
President Trump's tweet came after ex-UKIP leader Nigel Farage appeared on Fox And Friends, one of the president's favourite shows, talking about the weekend march.
"Dems want to greatly raise taxes for really bad and non-personal medical care," Mr Trump tweeted.
In response, Mr Hunt said that while he "disagreed with claims made on that march", no-one wanted "to live in a system where 28 million people have no cover".
He added: "NHS may have challenges but I'm proud to be from the country that invented universal coverage - where all get care, no matter the size of their bank balance."
Labour leader Jeremy Corbyn - the leader of the UK opposition - also hit back, saying: "People were marching because we love our NHS and hate what the Tories are doing to it."
He added: "Healthcare is a human right."
Donald Trump has said that he doesn't watch much television, claiming once he was too busy "reading documents".
Time and again, however, the president has praised and promoted specific Fox News programmes - particularly the cable network's morning offering, Fox and Friends.
Such was the case on Monday, as the president tweeted about the National Health Service shortly after former UKIP leader Nigel Farage appeared on Fox to criticise the NHS during a segment on a recent London protest march.
Healthcare reform, while a major topic of partisan debate in the US last year, has fallen off the radar recently.
Republicans tried, and failed, to roll back government regulation of the insurance industry instituted by Barack Obama.
In last week's State of the Union address, Mr Trump barely mentioned the topic.
When Fox News catches the president's attention, however, that can change with the flick of a Twitter finger.
NHS funding has been hotly debated in the UK as hospitals struggle to cope with the pressure on resources.
A panel set up by the Liberal Democrats - the fourth largest party in the Commons - has called for a ring-fenced tax to fund the service, saying an extra £4bn is needed for next year and an additional £2.5bn for both 2019 and 2020.