On a Monday morning in July, a gunshot rang out in the administration section of Milan's San Raffaele hospital. Seconds later, a frightened secretary entered the office of the institution's vice-president, Mario Cal, and found him lying in a pool of blood. Mr Cal clung briefly to life, but the Smith and Wesson revolver had done its job. Before long he died on one of his hospital's own operating tables.

The suicide of the hospital administrator went largely unnoticed outside Italy. But for the powers that be at the Vatican, it was more dreadful news. The global scandal over clerical paedophilia may be the story that dominates the headlines, but as the Vatican attempts to repair its stained reputation and mend diplomatic fences after spats with Beijing and Dublin, the death of Mario Cal was an ugly reminder of problems much nearer home.

Three hundred miles north of the capital, in Italy's second city, the battle is on to prevent a third body blow to the Vatican. This time cardinals are having to deal not with a moral abyss but a financial chasm – a shortfall of €1.5bn on the balance sheet of Milan's San Raffaele teaching hospital, an institute with links to the Vatican whose founder, Don Luigi Verzè, is a priest and a good friend of the city's most famous son, Italy's prime minister, Silvio Berlusconi.

The San Raffaele is highly regarded for the quality of its medical care and its research. But as Mr Cal's death made clear, it is in a crisis.

Thanks to bad investments and profligate expenses unrelated to medical care that would make a tycoon blush – including personal aircraft, hotels in Sardinia, and mango plantations in South America – the hospital is on the verge of collapse, something that was said to have been distressing Mr Cal greatly, and would have proved a huge embarrassment to the Vatican.

Underlining the gravity of the situation, Milan's chief prosecutor, Edmondo Bruti Liberati, announced on Friday that the organisation was under investigation for fraudulent bankruptcy.

In mid-September the Vatican proposed a €250m rescue package. But the proposals appear to have received a cool reception from the authorities. Now the Holy See has until 10 October to come with a better plan, or the institution is almost certain to be declared bankrupt at a hearing two days later.

Meanwhile, questions are being asked about the suicide and the nature of the institution in which it took place. How was the hospital able to build up such colossal debts? Who placed Mr Cal's gun in a bag away from the body before the police arrived? And why did a hospital administrator in one of Western Europe's safest major cities feel the need to keep a pistol in his desk?

Ongoing judicial investigations into the death and the hospital's disastrous finances may provide answers to these questions. Some observers are forthright with their suspicions, however, Vatican expert Paolo Flores D'Arcais, editor of the cultural magazine MicoMega, told The Independent: "I suspect we are talking about illegality and I hope that sooner or later that prosecutors will investigate it." Backing for his views emerged on Friday with news that Mr Liberati's "preliminary analysis" of files and paperwork in the office and at the private residence of Mr Cal has revealed "evidence of criminality" in the hospital's accounts. But the determination shown by the Vatican to save San Raffaele is also making waves.

The insititution appears so important that this summer Don Verzè was joined by Pope Benedict XVI's right-hand man Cardinal Tarcisio Bertone in the rescue operation. It wasn't as if the cardinal, the Vatican's head of state, didn't have enough on his plate with the seemingly endless child sex abuse scandal and its diplomatic consequences – as well as having to work in an atmosphere at St Peter's that is said to be poisonous.

But Cardinal Bertone, whose influence on the Vatican's purse strings, insiders say, has been achieved (and whom antipathy towards has been further fuelled) by placing friends and confidants in all the Holy See's key financial posts, was aware of the huge financial commitment the church was preparing to make in Milan, and felt obliged to step in. In mid-September, the Vatican Bank offered to stump up €250m to keep San Raffaele afloat, with some contributions coming from the businessman Vittorio Malcanzana, a friend of Cardinal Bertone. La Stampa reported last week that some at the Vatican doubt the wisdom of making such a huge financial commitment.

There has been talk of jobs being saved. But there may be other motives. Cardinal Bertone has said that he would like to create a larger centre of excellence, by merging the San Raffaele with other hospitals. Which begs the question: why is the Catholic Church so keen to be a major investor in the healthcare of Italy – a rich nation, with one of the highest life expectancies in the world – when poor countries lack a basic health infrastructure?

One suggestion is that medical care is profitable. "Nuns do much of the work nurses would do without paying them... and the church doesn't pay tax on the money it makes," says James Walston, a professor at the American University of Rome.

However, for Mr D'Arcais the Vatican's interest revolves around the exercise of power. "You have to understand the Vatican is not just about religion," he says. "It seeks to have greater power and influence, and what could be more important than having a pivotal influence on education, medical care and bioethics?"

With such influence over the hospital's purse strings, the Holy See would be able to deter activities such as embryonic stem cell research that it considers immoral. Academics at San Raffaele have said that freedom in teaching and research should be "non-negotiable". But some observers, including the Vatican Insider website, agree that were the Holy See to increase its financial stake, maintaining this position would prove an uphill struggle.