The survey, carried out by Ipsos MORI, also claimed that easing the workload on the NHS was one of the main reasons for people supporting the work.
It warned the eating out sector that "no concessions" were made for food consumed out of home – despite often being labelled as a "treat" by the pub and restaurant trade.
These figures come as Duncan Selbie, the chief executive of PHE, told the food industry that next year it will highlight where progress has not been made on sugar reduction - and further action from the government could take place if required.
He said that PHE will call on every sector of the food industry, in particular out of home outlets such as pubs, to "step up and accelerate its efforts".
The survey looked at the public's perception of obesity and the government's plans to reduce sugar and calories.
Other findings include:
• More than nine in 10 respondents think obesity is a problem in the UK
• 79 per cent believe it has a negative impact on the NHS
• Only cancer (47 per cent of respondents) and mental health (43 per cent) are seen as bigger health concerns than obesity (39 per cent)
• 90 per cent believe tackling obesity lies with individuals and families
• 80 per cent believe the food industry holds the greatest responsibility
• 72 per cent claimed the government should be ultimately responsible
Duncan said: "Obesity is the pandemic of modern times. Customers are saying they want faster progress...in particular in those businesses that have taken little or no action. We will be publicly reporting on these during 2019."
Dr Alison Tedstone, chief nutritionist at PHE, said: "Plans to improve the nation's diet are often described as 'nanny state' interference, but it's clear people want healthier food and they expect the industry to play their full part in this."