The spokeswoman, JoAnne Cavanaugh, refused to say how many workers had been suspended or granted exemptions due to medical or religious reasons.
Catholic Health said it was forced to postpone "a small number" of elective surgeries.
Erie County Medical Center in Buffalo suspended elective inpatient surgeries and had stopped accepting intensive-care patients from other hospitals as it prepares to fire hundreds of unvaccinated employees, a spokesman Peter Cutler said.
Mr Cutler said the decision to curtail some operations would inconvenience patients and hurt hospital finances.
Elective inpatient surgeries bring in about $1 million USD ($1.37 million AUD) per week, he said.
"We had to make a decision as to where we could temporarily make some changes so that we could ensure other areas of services are as little affected as possible," Mr Cutler said. "Financially, it's a big deal."
The inoculation push comes as President Joe Biden and other state and federal political leaders ratchet up pressure on unvaccinated Americans, some of whom object to mandates on religious or health grounds.
New York's state health department issued an order last month mandating that all healthcare workers receive at least their first COVID-19 shot by 27 September, triggering a rush by hospitals to get their employees inoculated.
Of the 43,000 employees at New York City's 11 public hospitals, about 5,000 were not vaccinated, Dr Mitchell Katz, head of NYC Health + Hospitals, said at the news conference with Mr de Blasio.
Dr Katz said 95 per cent of nurses were vaccinated and all the group's facilities were "open and fully functional" on Monday.
On Saturday, New York Governor Kathy Hochul said she was considering employing the National Guard and out-of-state medical workers to fill staffing shortages, with 16 per cent of the state's 450,000 hospital staff not fully vaccinated.
Healthcare workers who are fired for refusing to get vaccinated will not be eligible for unemployment insurance unless they are able to provide a valid doctor-approved request for medical accommodation, Ms Hochul's office said.
A federal judge in Albany temporarily ordered New York state officials to allow religious exemptions for the state-imposed vaccine mandate on healthcare workers.
Separately, a federal appeals court on Monday ruled that New York City can order all teachers and staff to get the vaccine, reversing a previous decision that had put the mandate on hold for educators.