Neither Russia nor Ukraine are members of the ICC, but Kyiv has accepted the court’s jurisdiction.
Mr Zelensky has also called for the West to consider a no-fly zone over Ukraine – something Washington so far ruled out over fears it could draw the US into a direct conflict with Russia.
Videos shared on social media showed rockets landing in Kharkiv, in what some defence analysts described as typical of a cluster munition strike on a dense urban area.
Amnesty International called for an investigation into the assault on Kharkiv as a possible war crime.
Russia has previously denied targeting residential areas.
UK Prime Minister Boris Johnson accused Russia of “barbaric Russian airstrikes against innocent civilians, including children” and pledged more military aid for Ukraine.
Established in 2002, the ICC investigates and prosecutes genocide, war crimes and crimes against humanity.
Any Russian nationals accused of committing war crimes, crimes against humanity or genocide in Ukraine, would have to be extradited by the Kremlin before standing trial in The Hague.