ALBANY, N.Y.—Lawmakers and unions criticized Gov. Andrew Cuomo for pushing ahead with a plan to reduce Medicaid spending as part of the state budget due Tuesday even as New York responds to the novel coronavirus pandemic.

Several Democrats who control the state Senate as well as labor and progressive groups warned that cuts to the program, which provides health care to more than 6 million New Yorkers, would be devastating. They urged Mr. Cuomo to accept federal money approved earlier this month and to tax the wealthy rather...