The figure had been a central part of the campaign's "let's take back control" message, with suggestions that the money could instead be used to fund state healthcare.
"And that's what we're doing because we had a referendum campaign past year, a lot of strong language was used, it got quite bitter at times".
"It is a clear misuse of official statistics", Sir David added in his brief note.
In Mr Johnson's reply, he wrote: "I must say that I was surprised and disappointed by your letter of today, since it was based on what appeared to be a wilful distortion of the text of my article".
Remainers said the figure - printed on the side of a Leave campaign bus alongside the words, "Let's fund our NHS instead" - failed to take into account the full economic impact of being part of the European Union, and of the rebate.
The Independent: Brexit: Theresa May warned over £2bn worth of UK exports to Canada at risk as she flies to Ottawa
The rebuke piled pressure on Mr Johnson, who is already accused by some in his party of seeking to undermine Theresa May's leadership and approach to Brexit.
A senior minister who backed Remain told The Sunday Telegraph, meanwhile, Mr Johnson "needs to go and do something else" if he "can't settle" into his role as foreign secretary, while a former minister said he was "sailing within an inch of being thrown out of the Government".
What is beyond doubt is that, upon withdrawal, we will have complete discretion over the £350m per week and that huge sums will indeed will be available for public spending on priorities such as the NHS.
The Labour MP Chuka Umunna, a leading supporter of Open Britain, said: "Yet again Boris's outright lying has been exposed by Britain's statistics watchdog".
Later, after the motives and timing of his intervention were criticised, Mr Johnson insisted in a tweet linked to his article that he was "looking forward to PM's Florence speech". In a barely coded threat to May's position, Johnson's allies warned she would be on a "collision course" unless she listens to him when she makes a speech in Florence on Friday about the negotiations.
Interior minister Amber Rudd said it was "absolutely fine" for the foreign secretary to intervene publicly but that she did not want him managing the Brexit process.
Controversially, Mr Johnson says that foreigners could be prevented from buying property in the UK. "I am going to make sure, as far as I and the rest of the cabinet is concerned, we help her do that", she told BBC One's The Andrew Marr Show.