Of the 23 injured, 21 are in hospital and three are in a serious condition.
Two boys aged three and six were also among the wounded, governor Serhiy Lysak said.
Russian strikes on Ukraine have intensified in recent weeks ahead of an expected Ukrainian counteroffensive.
Mr Zelensky posted a video of the damaged clinic that showed firefighters at the scene and smoke billowing from the building.
"Russian terrorists once again confirm their status of fighters against everything humane and honest," he said.