Ban's remarks came after pro-Kremlin rebels announced a major new offensive and heavy rocket fire killed at least 30 people and wounded 97 in Mariupol, a strategic port linking separatist territory with Russian-occupied Crimea.
Ban "strongly condemns today's rocket attack on the city of Mariupol," a statement from his spokesman read.
"He notes that rockets appear to have been launched indiscriminately into civilian areas, which would constitute a violation of international humanitarian law."
The United Nations chief also condemned Friday's "unilateral withdrawal from the ceasefire by rebel leadership, and particularly their provocative statements about claiming further territory."
He said the withdrawal constituted a violation of commitments under September's Minsk accords that ordered the withdrawal of heavy weapons and a truce.
"Ukraine's peace, territorial integrity and stability, intrinsically linked to that of the broader region, must be urgently restored," the statement added.
Alexandre ZakharchenkoZakharchenko, leader of the self-declared Donetsk republic, announced earlier the launch of an offensive on Kiev-controlled Mariupol.
The Ban statement did not say who was responsible for the rocket fire.
Meanwhile, the Security Council tried to put together a statement on Mariupol, at Britain's urging. But Russia, often accused of being close politically and militarily to the separatists, blocked the effort, western diplomats said.
Ban "issues a strong statement on Mariupol attack but Russia blocks similar statement by the UN Security Council," British Ambassador Mark Lyall Grant tweeted.
Lithuanian Ambassador Raimonda Murmokaite also took to Twitter to say that "'DPR' chief Zakharchenko admits onslaught on Mariupol but Russia continues to shield its proxy, so no Security Council statement."
Speaking privately, a Western diplomat said "it is disappointing that it has not been possible to agree a UN Security Council statement on the shelling today as the Russian federation were against including a reference to a public statement on an offensive on Mariupol which had been made by a separatist leader."
Since the crisis broke out in Ukraine, the council has held more than two dozen sessions on the issue, without being able to reach consensus. Moscow denies involvement with the separatists and as a permanent council member, Russia has veto power.