The Harley Medical Group, which fitted PIP breast implants to almost 14,000 British women, has said it will not replace them free of charge.
The private firm fitted more of the implants than any other UK cosmetic surgery firm, claiming replacing the banned implants would put it out of business.
It follows days of delay from the company in giving clear advice to women affected by the scandal.
Harley Medical Group's chairman Mel Braham told the BBC the firm had neither the resources, the surgeons, nor the operating facilities needed to do the surgery.
He said the government must accept "moral responsibility" for replacing the PIP implants.
He added: "We're only sitting here today because the Medicines and Healthcare products Regulatory Agency (MHRA), their own regulatory authority, has approved these implants and obviously hasn't done their proper checking.
"We're an innocent victim like everyone else, we're attempting to do our best for our patients."
He said the Harley Medical Group would supply the NHS with replacement implants if that was the patient's decision.
But he said: "We can't take on this whole thing on our own, especially when it wasn't our fault."
Other leading providers including BMI Healthcare, Nuffield Health and Spire have agreed to offer free removal of the PIP implants.
The Hospital Group has said it will only pay to replace PIP implants that have ruptured.
Another private company, Transform, has also said it will not replace the implants free of charge.
Around 40,000 British women received PIP implants manufactured by the now closed French company Poly Implant Prostheses (PIP).
The implants were filled with non-medical grade silicone intended for use in mattresses.
The government has said anxious patients who had their surgery on the NHS will be able to have the implants removed and replaced free of charge.
The NHS will also pay to remove, but not replace, implants if a private clinic refuses or no longer exists.
Yesterday, the Welsh government said it would pay to both remove and replace breast implants from women who had them fitted by a private clinic.
Earlier on Wednesday, Health Secretary Andrew Lansley reiterated the government's view that private clinics have a moral duty to look after their patients.
He also announced further investigations into the scandal, including looking at the EU approval of the devices which resulted in them receiving a CE mark. He also reiterated the views of an expert panel convened to look at the issue, saying there is no "causal link" between the implants and breast cancer.
The panel found it was undeniably the case that the implants are made up of non-medical-grade silicone and should not have been implanted in women in the first place, but was unable to establish if the rupture rate is higher for PIP implants than for others.
Lansley said the events "highlight the need to ensure the safety of people having cosmetic interventions".
"A review for this purpose will be led by the minister for quality... It will investigate and report rapidly. The blame for what happened lies with PIP, but this review will enable us to learn lessons to improve future regulatory effectiveness."
Lansley repeated the pledge for the Care Quality Commission to conduct a swift review of private clinics. This will look at evidence of compliance, patient safety and clinical quality, and the information and support given to patients.
A review has already been set up into the regulation and governance of the entire cosmetic surgery sector, led by Professor Sir Bruce Keogh, and will include treatments such as cosmetic surgery and dermal fillers.
"The review will consider issues of governance, data quality, record keeping and surveillance, as well as ensuring that proper information is provided to secure patients' informed consent," the health secretary said.
In France, the government has told 30,000 women they should have the implants removed, while the Czech and German authorities have recommended that women should also have them taken out.