He made the pledge during this summer's leadership contest against Liz Truss.
Mr Sunak had argued it was "not right" some patients were failing to turn up, taking slots from those in need.
But a No 10 spokeswoman said that, after "listening to GPs", the government decided it was "not the right time" for the policy.
She added that Mr Sunak wanted to deliver "a stronger NHS and the sentiment remains that people should not be missing their appointments and taking up NHS time".
Critics of the proposal included doctors' union the British Medical Association (BMA), which said it would "make matters worse" and threaten the NHS's principle of free care at the point of need.
The BMA welcomed the decision to scrap the plan and said it "cannot be brought back to the table later down the line".
It said "punishing" patients was not the answer to the many reasons people miss appointments, and the policy would "only deter the most vulnerable from seeking the help they need, worsen health inequalities, and ultimately undermine the essential trust between doctor and patient".
Instead, the BMA said the government must work to improve pay and other conditions to ensure the NHS has safe levels of staff.
Reflecting a similar view, the Royal College of GPs said the plans would have disadvantaged some of the NHS's most vulnerable patients.
Chairman of the college Prof Martin Marshall said the move would "simply have been tinkering at the edges given the scale of the crisis facing GPs and our teams".
Outlining the policy in July, Mr Sunak said patients would be given the "benefit of the doubt" the first time they missed an appointment without providing sufficient notice, but further absences would incur a £10 charge.
Fines would also be waived in exceptional circumstances, such as if a patient had an emergency.
He explained the system would be "temporary" as backlogs caused by the pandemic were cleared.
But he gave few details of the how the system would work, writing in the Sunday Telegraph: "If we have people who are now showing up and taking those slots away from people who need [them], that's not right.
"I'm all for a healthcare system that's free at the point of use, but not one that's free at the point of misuse."
Asked if other pledges on the NHS made during the summer leadership campaign still stood, she said Mr Sunak's "initial ideas" over summer would be discussed with cabinet ministers and any announcements would be "set out in due course".
Mr Sunak had also pledged to eliminate one-year waiting times by September 2024, and get the number of people waiting for non-urgent treatment in England falling by next year.
He pledged to do this by boosting the number of "diagnostic hubs" outside hospitals, including by repurposing empty High Street shops. He also pledged to reform dentists' NHS contract, and ring fence the annual £3bn NHS dentistry budget.
The decision came as Mr Sunak met patients and staff at Croydon University Hospital in his first visit as prime minister.
A patient was filmed telling the prime minister he needed to "try harder" to pay NHS staff more.
Mr Sunak replied "he would take that away" following the exchange.
During the summer the Conservative Party membership chose Ms Truss over Mr Sunak, but this week he succeeded her by winning enough nominations from Tory MPs on Monday following her resignation.
WATCH: Catherine Poole, aged 77, tells Rishi Sunak NHS staff should be paid more