The dispute is dividing hospitals, where most workers are vaccinated and want their colleagues to be. The nurses’ union supports the mandate — some 95 percent of members are already vaccinated — even as some members complain its rollout was too rushed. But unions representing support workers, including nurses’ aides, orderlies, cafeteria workers and others, have opposed it. If many of those workers leave or are fired, their duties could fall to already taxed nurses.

The disagreement is also testing government’s power to mandate compliance with public-health measures; New York’s mandate and the state’s refusal to allow religious exemptions are the subject of at least two lawsuits, including one by Ms. Conrad and five other plaintiffs.

Still, staff members choosing to exit their jobs because of the mandate could also create immediate practical challenges: Many nurses and other health workers are burned out or traumatized from the pandemic’s strain; others have been lured by high salaries to become “travel nurses,” crisscrossing the country to fill emergency staffing gaps.

On Sunday at the Christian Cultural Center in Brooklyn, Gov. Kathy Hochul pushed back hard against the idea of religious exemptions to vaccination, urging worshipers to be “apostles” for the vaccine in order to “keep more people alive.”

“God did answer our prayers,” she told the congregation. “He made the smartest men and women — the scientists, the doctors, the researchers — he made them come up with a vaccine. That is from God to us and we must say, ‘Thank you, God, thank you!’”

“There are a lot of people out there who aren’t listening to God and what God wants,” she said as a gold necklace spelling “Vaxed” glinted from her chest.