In his post, which has since been removed, Mr Bradley wrote: “That’s what FSM (free school meal) vouchers in the summer effectively did...”
His comments came after hundreds of businesses and many local councils announced they will feed poor pupils this half-term if the government will not.
England footballer Mr Rashford had launched a campaign for free school meals during the break, after he convinced the government to make a U-turn on summer holiday meals earlier this year.
After cafes, pubs and restaurants said they would provide the meals, one Tory MP told them she hoped they “will not be seeking any further government support”.
Selaine Saxby, the Tory MP for North Devon and a former teacher, also said on social media she was “delighted our local businesses have bounced back so much after lockdown they are able to give away food for free”.
In an interview with BBC Breakfast, Mr Bradley claimed his tweet had been taken out of context, though he did not explain how. He also criticised the government’s communication around this week’s vote.
The MP for Mansfield said “not everything has to come from central government” and local communities also have a role to play.
He said: “It wasn’t a vote on Wednesday of ‘do we or do we not help kids’, what we’ve done is put a billion pounds on to local government this week to help them to target the most vulnerable families and help perhaps more precisely and more directly those most in need.
“My point really is only that not everything has to come from central government. I think Marcus’s campaign has been brilliant in rallying round communities who also have a role to play in supporting people and I think that’s really positive.”
On his controversial tweet, he said: “I was merely making the point that there are kids who live in really chaotic situations, really difficult lives, where actually giving them an unrestricted voucher to spend on whatever isn’t helpful.”
Campaigners insist the vouchers are not unrestricted, and cannot be used to pay for certain items.
Labour’s deputy leader Angela Rayner said: “Notwithstanding the fact that the vouchers in summer could only be used to purchase food, this stigmatisation of working class families is disgraceful and disgusting.”
Shadow children’s minister Tulip Siddiq has written to the co-chair of the Conservatives Amanda Milling calling on her to distance her party from the comments, and adding: “I respectfully ask you to request an apology from Mr Bradley to the millions of children from lower income households who benefit from free school meal support.”
The row adds to mounting pressure on Boris Johnson to perform a U-turn on the issue.
Conservative MP Robert Halfon, who rebelled and voted to extend free school meals during the holidays, called on the prime minister to meet Mr Rashford.
The chair of the Commons education select committee said the move was a “no-brainer”, adding: “It may be that they (No 10) don’t agree with everything that Marcus Rashford is proposing, but it would give us a chance to come up with a long-term plan to combat child food hunger once and for all.”
Earlier a government minister defended the decision not to extend free school meals through the English school holidays.
Civil society minister Baroness Barran told BBC Radio 4’s Today programme: “The government has made over £200bn available to protect people’s incomes and people’s livelihoods.
“We have worked very closely with charities who operate food banks across the country. There are different approaches to how we do this but we have used all the levers possible to try to make sure that people are safe and well as we go forward.
“We have provided more money through universal credit, we have provided more money to local authorities and we believe that is a better way of approaching this problem.”