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.
Earlier, Mr Lysak said the region came under a "mass attack...with missiles and drones" on Thursday night.
"It was a very difficult night. It was loud," he said. "Dnipro has suffered."
At the scene, fire crews were sawing down trees to get a mounted hose closer to the flames which had engulfed the large, three story building.
Meanwhile, amidst the rubble, rescue teams were searching for two missing people.
Ukrainian authorities said they shot down 17 missiles and 31 drones launched from Russia overnight.
Several drones and missiles hit targets in Dnipro and the eastern city of Kharkiv, including an oil depot.
Ukraine's capital, Kyiv, was also targeted and officials said fragments of intercepted drones fell on the roof of a shopping centre, while a house and several cars were damaged.
In Russia, a blast damaged a residential and office building in the southern city of Krasnodar, east of Crimea, on Friday morning.
The region's governor, Veniamin Kondratyev, said it was caused by two Ukrainian drones: "There is some damage to buildings, but critical infrastructure was not damaged. And most importantly, there were no casualties."
Russia's Belgorod region, which was the scene of an unprecedented incursion from Ukrainian territory earlier this week, was also hit overnight. The village of Kozinka was struck more than 130 times, according to its governor Vyacheslav Gladkov.
Mr Gladkov said one woman was injured. He said the Graivoron district, where the incursion took place, was subject to the worst attacks.