The alert, sent to medical providers and other subscribers, went on to warn that the consequences of relying on potentially false results may lead to “providing patients incorrect guidance on preventive interventions like physical distancing or protective equipment.”

Dr. Daskalakis, early in the outbreak, had been a strong voice arguing for social distancing measures and urging Mayor Bill de Blasio to close schools in New York City. When Mr. de Blasio resisted doing so, Dr. Daskalakis threatened to resign, a city official said. City Hall has said the mayor was never told of the threat.

Patrick Gallahue, a spokesman for the city health department, said the alert referred to “unvetted tests outside of a lab setting — which is not what the state is doing.”

He added that there “isn’t any daylight here” between the city and state.

Dr. Michael Osterholm, an infectious disease expert at the University of Minnesota, praised the overall intent of New York’s study, but said the results in this case probably skew to a higher estimate than is real because a survey of grocery store shoppers in a pandemic would not be representative.

The sampling may disproportionately include those who have either already had the illness, or those who naturally tend to go out more and so are more likely to be exposed to the virus, he said. It would miss children, teenagers and older adults who may be sheltering in place.

“It’s not a criticism. It’s more of a comment that we have to be careful about interpreting supermarket customers as a representative sample when the state was in lockdown,” he said.

State officials said the test had been calibrated to err on the side of producing false negatives — to miss some who may have antibodies — rather than false positives, which would suggest a person had coronavirus antibodies when they did not.