“Dottie, how are you?” he asks a woman in a purple robe who is suffering from obstructive lung disease. “I’m taking a look at your X-ray right now, and it looks pretty good.” She sits up in her bed, breathing through a tracheostomy tube. “I’ll check in with you later,” he promises.

Hayes’s pace might be called breakneck if he was moving from bed to bed. But he’s remotely monitoring patients at several hospitals from an “e-ICU’’ outpost in Westwood operated by Steward Health Care System. Caregivers at the site watch video screens showing patients in hospital ICUs from Methuen to Fall River. It brings extra sets of eyes to intensive care while letting Steward keep its staffing lean.

Like everything else about for-profit Steward — robotic surgery, fixed-rate insurance contracts, managers working with patients to prevent hospital readmissions — the e-ICU is focused on innovation, efficiency, and finding ways to save money.

It’s a formula that has been reshaping the way business is done in the state’s health care industry ever since Steward was formed by a New York buyout firm in 2010 to take over the struggling Caritas Christi Health Care chain. But whether the makeover will achieve its most important goals — making medical care less expensive in Massachusetts and making a profit for Steward — remains an open question. What is certain is that Steward has become a force in this critical industry.

“There’s a bubbling caldron of change going on in Massachusetts health care, and Steward is the single biggest part of it right now,” said James Roosevelt Jr. , chief executive of Tufts Health Plan, which has teamed up with Steward to sell a limited-network insurance product to small companies located near Steward hospitals.

Just more than two years after state officials gave Steward control of the six Caritas hospitals — including Boston’s St. Elizabeth’s Medical Center and Carney Hospital — the new system has lost tens of millions of dollars. But it has also assembled a formidable network of 11 hospitals with nearly 2,100 beds, second only to the giant Partners HealthCare System. Steward says it is forging a lower-cost “community care” model, drawing patients from expensive Boston teaching hospitals. It is forcing every other player in the medical business to reassess its own strategy.

A report released by state Attorney General Martha Coakley’s office Wednesday said Steward posted a $14.6 million operating loss for its 2011 fiscal year. It piled up a total deficit of $56.9 million that included one-time accounting charges related to the costs of acquiring and renovating its hospitals. The report said Steward has complied with conditions imposed by Coakley when she recommended approval of its bid to buy the Caritas hospitals, but it also noted the system was highly leveraged. Steward borrowed nearly $100 million in its first year.

Caritas, which had been affiliated with the Roman Catholic Archdiocese of Boston, had threatened to close some of its money-losing hospitals before the Steward deal gave it a backer with deep pockets — a factor that influenced Coakley’s recommendation.

For its partners, rivals, and regulators, the face of Steward is its chief executive, Ralph de la Torre, a former cardiac surgeon and Caritas president who engineered the deal with private equity firm Cerberus Capital Management and then signed on to run Steward. De la Torre brims with confidence about the chain’s future, even though it has yet to turn a profit.

“We have a pretty complete business plan,” said de la Torre, who insisted the company expected to lose money its first year. “By every measure, we’re actually ahead of the plan.”

The system’s rapid rise has forced other hospitals to figure out whether they can afford to keep going it alone or need to connect with a stronger partner. Many are choosing the latter route — just last month, Jordan Hospital in Plymouth signed a letter of intent to be taken over by Boston’s Beth Israel Deaconess Medical Center.

Steward “has capital, and capital is the name of the game,” said Donald J. Thieme, executive director of the Massachusetts Council of Community Hospitals. “The new model seems to be bigger is better if you’re going to survive. Every hospital has to think about its place in the future.”