The shadow health secretary warned that some NHS hospitals were planning to double the number of private patients they treat under new freedoms.
In a speech to Labour's conference, he said the coalition risked destroying the character of the NHS that David Cameron had promised to protect.
Labour has not ruled out all competition to provide NHS services.
In the party's 2010 manifesto, it promised patients the right to go private if they were still waiting for an operation at 18 weeks and an "active role" for the independent sector in treating NHS patients.
By Nick TriggleHealth correspondent, BBC News
The role of the private sector in the NHS is a thorny issue.
Ever since the health service was created it has bought in expertise from outside when necessary.
Today it is estimated about Â£1 in every Â£20 spent on NHS care goes to non-NHS providers.
Some of this pays for charities to help in areas such as cancer and mental health care, while some is handed over to private health firms to carry out knee and hip replacements.
The last 20 years or so has been marked by a desire by politicians from all sides to use the market to make the NHS more efficient.
Labour under Tony Blair expanded this under the patient choice initiative.
That allowed patients to choose  non-emergency hospital treatment from an approved list of providers, including many from the private sector.
This set the wheels in motion for a greater role for private firms in the community sector - one of the issues Andy Burnham is warning about now.
Either way, the genie was out of the bottle long ago.
But the party believes the scale and pace of "marketisation" under the coalition threatens the foundations of the health service.
In particular, Mr Burnham said he feared the new freedom for hospitals to earn 49% of their income from private work would "damage the character and culture" of the NHS and take it closer to an American model.
In his speech, Mr Burnham said: "We can save the NHS without another structural re-organisation.
"I've never had any objection to involving doctors in commissioning. It's the creation of a full-blown market I can't accept. So I don't need new organisations. I will simply ask those I inherit to work differently.
"Not hospital against hospital or doctor against doctor. But working together, putting patients before profits."
The shadow health secretary also called for the NHS and local authorities to work closer together to improve care for the elderly and for "full integration" of health and social care services to be actively considered.
"If we can find a better solution to paying for care, one day we might be able to replace the cruel "dementia taxes" we have at the moment and build a system meeting all of a person's needs - mental, physical, social - rooted in NHS values," he added.
Andy Burnham has vowed to reverse the "rapid" privatisation of NHS hospitals in England if Labour wins power.