The shift by Mr. Cuomo, a Democrat who had long resisted legalizing medical marijuana, comes as other states are taking increasingly liberal positions on it — most notably Colorado, where thousands have flocked to buy the drug for recreational use since it became legal on Jan. 1.

Mr. Cuomo’s plan will be far more restrictive than the laws in Colorado or California, where medical marijuana is available to people with conditions as mild as backaches. It will allow just 20 hospitals across the state to prescribe marijuana to patients with cancer, glaucoma or other diseases that meet standards to be set by the New York State Department of Health.