A Royal College of Surgeons (RCS) report said the situation was "dangerous" and that children were also suffering due to operation delays.
Cynon Valley MP Ann Clwyd claimed it was the Welsh equivalent of the Stafford Hospital scandal.
Hospital chiefs refuted the comparison and said problems were being tackled.
Adam Cairns, chief executive of Cardiff and Vale University Health Board (CVUHB), told BBC Wales Today that 12 patients had died over a 15 month period while on the cardiac waiting list.
He said he could not be certain whether any one of those deaths was avoidable but shortening the length of time people waited reduced the risk.
The report by the RCS on UHW - the biggest hospital in Wales - followed a visit of its surgical departments by the college's Professional Affairs Board for Wales (PAB) in April.
The report said there was "universal consensus" amongst the clinicians that services at the hospital were "dangerous" and of "poor quality".
"Urgent attention" by Cardiff and Vale University Health Board (CVUHB) was needed to address the issues, it said.
An action plan has been agreed with the health board and will be reviewed in the autumn.
Ms Clwyd, who is conducting a review into complaints by hospital patients in England, told BBC Radio Wales that a public inquiry into the hospital should be chaired by someone from outside Wales.
She pointed to UHW having the highest mortality rate of any hospital in Wales adding that UHW accounted for the "vast majority" of complaints she had received from Wales.
"I think it's very serious - I think it's the equivalent of Wales' Mid Staffs," Ms Clwyd said.
"It was mortality rates in Mid Staffs that first raised the alarm [there] and because of that there was an inquiry."
The Francis inquiry into failings at Mid Staffordshire NHS Foundation Trust between 2005 and 2008 showed there were between 400 and 1,200 more deaths at Stafford Hospital than would have been expected.
The report argued for "fundamental change" in the culture of the NHS to make sure patients were put first.
It prompted a separate review of 14 NHS hospitals in England with high mortality rates.
Ms Clwyd added: "This is a shocking set of events and the report from the Royal College of Surgeons, who as you know are fairly conservative in their approach to things, has made these severe criticisms of the University Hospital of Wales.
"And I think both the chairman and the chief executive should consider their own positions.
"They're responsible for running that hospital, they've got questions to answer. And in my view, they have not answered them, and the only way this will happen is through a proper inquiry like the Francis inquiry into Mid Staffs."
Cardiff and Vale Health Board said revised figures for death rates at the hospital were significantly better than those originally reported in March adding that it needed "to make further improvements including targeted audits of clinical specialities where we have concerns and continued improvement in our data quality and coding activities".
The RCS report said concerns included:
The single most common complaint from the hospital's surgeons was the inability to admit patients for scheduled or elective surgery.
They reported that more than 2,000 operations were either not scheduled due to a lack of beds or cancelled in the first three months of this year.
"I think our colleagues were telling us that they believed there was significant risk as a consequence of their inability to get patients in to have their operations in a timely way," said Colin Ferguson, director of the PAB.
"The reasons for that are obviously complex and relate to the whole system of care within the hospital. They relate to emergency medicine, they relate to the admissions coming through the A&E department.
"They were very frustrated. As a group they were expressing to us very clearly that they thought the current situation was unsustainable and something had to be done about it."
The report said doctors also believed the health board was reducing its scheduled surgery in order to reduce costs "to meet end of year financial targets".
An increase rise in waiting lists, according to the report, meant that patients "were clearly coming to harm".
Mr Cairns said the pressures over the last winter were unprecedented in his 30 years experience but he accepted the situation earlier in the year was cause for concern.
"I'm on record already having apologised to our patients," he said.
"It is not a position that we can take any pride in - we need to do better.
"It is fair to say that this organisation was struggling, like many others, to cope with the pressures," he added.
"The circumstances that we found ourselves in, I think, are unacceptable, and we've got to make sure that in the future that we have a much better response."
Mr Cairns said steps had been taken to protect surgical beds for children's services, and to find more resources for cardiac treatment.
"Some of the things that we are already doing are showing some improvements," insisted the chief executive.
"The bottom line here is we can, and we will, do better."
Later, Mr Cairns issued a statement saying: "We are extremely disappointed that Ms Clwyd has chosen to liken the RCS report to what happened at Mid Staffordshire Hospital and would strongly refute her comparison."
He said the health board had "an open and collaborative culture" and was taking steps to rectify problems quickly.
The patients' watchdog Cardiff and Vale Community Health Council also dismissed the Mid Staffordshire comparison, saying it "has major concerns on how this analogy will impact on patients' confidence, who are either undergoing or awaiting treatment in Cardiff".
The RCS has also taken the step of passing its report to Mark Drakeford, Health Minister in the Labour-run Welsh government, which led to a meeting in June.
"The health minister was very concerned to read the Royal College of Surgeons report and the risks to quality and safety it highlighted," said a Welsh government spokesperson.
It led to an action plan being drawn up by the health board, NHS Wales and Welsh Health Specialised Services.
The RCS said it will now make a return visit to the hospital and health board in September to review the situation.
An MP has called for an inquiry after surgeons said patients had died waiting for heart surgery at the University Hospital of Wales (UHW) in Cardiff.