Garry Pettigrew, Managing Director of Healthcare Environmental Services (HES), said it was not true that amputated limbs and other tissue were among the refuse that built up.
Fifteen NHS trusts in Yorkshire and Humber ditched their waste contracts with the company. The Environment launched enforcement action and placed restrictions on the HES site in Normanton.
Mr Pettigrew told the BBC that "anatomical waste" was always stored securely and prioritised for destruction.
And he repeated his claim, made when the issue became public last Friday, that a "lack of incineration capacity" was behind the problem.
He told the BBC: "Every single part that people are referring to there is dealt with securely, professionally, and any anatomical waste would be stored in fridges and at the same time prioritised for outward bound."
On Tuesday, health minister Stephen Barclay told MPs more than 3.5 tonnes of human body parts was stockpiled at four sites by HES, which collected Â£31m last year to burn waste, but "just 1.1% of this clinical waste is anatomical".
Some of its contracts have now been terminated after the Environment Agency said on October 5 that HES had been found to be in breach of permits at four of its six sites in England which deal with clinical waste and a criminal investigation has been launched.
As part of its enforcement activity, it has partially suspended the company's permit at one of its sites, which will prevent it from accepting any more incinerator-only waste.
The Scottish Environment Protection Agency (Sepa) also confirmed that it issued enforcement notices at sites in Dundee and Shotts last month, where its officers are conducting "ongoing monitoring".
The Department of Health and Social Care (DHSC) said there was "absolutely no risk" to public health.
Mr Pettigrew told the BBC on Friday that the fault was down to a fall in capacity at incinerators.
He said: "We've had this contract since 2010. If you go back to the articles in 2010, this was the jewel in the crown of NHS England, that we had saved them Â£30m for awarding this contract to us.
"For the last eight years we have done this contract and never seen the situation we are in now."
The Environment Agency has previously rejected this, saying "there is industry-wide agreement that overall there is sufficient incineration capacity".
Trusts in Leeds, York, Sheffield, Bradford and North Lincolnshire were among those served by the HES Normanton site.
From Monday morning, facilities management company Mitie took over their waste disposal services.