He denounced the attack as a crime against humanity, and the Ukrainian Defence Ministry called it a serious war crime under the Geneva Conventions, which set out how soldiers and civilians should be treated in war.
Video footage showed a devastated building with smoke pouring out of it and rescue workers looking on. Much of the upper floor of what appeared to be a three-storey building had been badly damaged, as had cars parked nearby.
“Another (Russian) missile attack, another crime against humanity,” Zelenskyy wrote on Twitter.
“The buildings of a psychological clinic and a veterinary clinic in the city of Dnipro were destroyed. As of now, one person was killed and 15 were wounded.”
Russia has denied repeated accusations that its soldiers have committed war crimes in Ukraine.
Ukrainian officials said earlier on Friday that air defences had shot down 10 missiles and more than 20 drones launched by Russia in overnight attacks on the capital Kyiv, Dnipro and eastern regions.
Reuters was unable to verify the reports.
Russia, which launched its full-scale invasion of Ukraine 15 months ago, did not immediately comment on the reported air strikes but said Ukraine had struck two regions in southern Russia with a rocket and a drone.