Stewart Jackson, who served as special adviser to Mr Davis in the Department for Exiting the EU (DExEU) until earlier this year, has also been branded a “nasty man” after the tweet.
On Monday night, Mr Jackson responded to a fellow Twitter user who posted a photo of the child, alongside another image of two children wearing “B******* to Brexit” hats.
Alongside the photos, Anthony Hobley revealed how his stepson had recently had an operation at Great Ormond Street Hospital in central London and was “gutted” he couldn’t be at a demonstration in support of a referendum on the final Brexit deal.
On Saturday, an estimated 700,000 marched in London as part of the “People’s Vote” campaign.
Mr Hobley wrote: “My stepson had an operation yesterday @GreatOrmondSt. He’s incredibly brave but gutted he can’t be at the #PeoplesVoteMarch today with his brothers & sisters.
“You can see he’ll be there in spirit & his brothers are saluting him with a dab! #PeoplesVote #peoplesvotemarch”
In response to the tweet, which he now appears to have deleted, Mr Jackson said: “What a pathetic cretin.”
Mr Hobley, from Surrey, is now demanding an apology from Mr Jackson.
“Comments like these from senior people in political life set the tone of political debate in our country and I believe harm politics in our country,” he said.
“Whatever our views on Brexit we need to talk about the problems of Brexit and the case for a People’s Vote by debating the facts and issues, not with schoolyard abuse.
“I don’t believe people honestly worried about the future of our country should be subject to trolling by senior political figures, especially not when it involves children.”
Mr Jackson, a committed Leave supporter who was a Conservative MP for Peterborough between 2005 and 2017, was also criticised by prominent Labour and Tory figures.
Ex-Tory minister Anna Soubry tweeted: “When any scrap of humanity is overwhelmed by ideology @BrexitStewart is a bitter man gone badly wrong.”
Her fellow Conservative backbencher Heidi Allen wrote: “What a pathetic, unkind and nasty man.”
Labour MP David Lammy claimed Mr Jackson had been “rattled by the nearly three quarters of a million” who marched on Saturday, adding: “There’s only one pathetic cretin to see here.”
Fellow Labour MP Wes Streeting told Mr Jackson the tweet was “beneath you”.
“You should delete it and issue a grovelling apology,” Mr Streeting added.
Mr Jackson suggested his comment was aimed at Mr Hobley rather than the child, replying to Mr Streeting: “What sort of individual invades their sick child’s privacy at a sensitive time to make a political point about the so called People’s Vote? Really. Awful.”
He also told website Politico: “I think it’s awful that people with extreme views on Remain like this parent should invade a sick child’s privacy to make a political point. He should be ashamed of himself.”
Mr Jackson left DExEU alongside Mr Davis in July, when the former Brexit secretary quit in protest at the prime minister’s Chequers plan.
Sky News has contacted Mr Jackson for comment.