Ban said: "While recognising Israel's right to defend itself, I condemn the disproportionate and excessive use of force that has killed and injured so many civilians, including children ... I call on Israel to cease such attacks."
"I condemn Palestinian rocket attacks and call for the immediate cessation of such acts of terrorism."
In protest against the Israeli offensive, Mahmoud Abbas, the Palestinian president, has formally suspended all contacts with Israel.
"The negotiations are suspended, as are all contacts on all levels, because in light of the Israeli aggression such communication has no meaning," Nabil Abu Rudeina, an Abbas spokesman, said in a statement.
However, there is no sign that the Israeli government is ready to call off an offensive that took troops deeper into Gaza on Saturday and in larger numbers than at any time since Israel ended a military occupation of Gaza in 2005.
West Bank clashes
On Sunday, fresh Israeli strikes killed three Palestinians in northern Gaza.
Earlier, witnesses said Israeli aircraft attacked the office of Ismail Haniya, the ousted Palestinian prime minister, in Gaza City.
As Gazans mourned the dead, clashes between Palestinians and Israeli troops erupted in the Shufat refugee camp in occupied East Jerusalem, Hebron, and Belin as well as other areas across the West Bank.
Hundreds of West Bank residents, holding Hamas and Fatah flags, also staged rallies and appealed for national unity.
In a related development, Egypt has agreed to open the Rafah border crossing to let injured Palestinians receive medical treatment, Al Jazeera’s correspondent reported.
Daniel Carmon, a senior Israeli diplomat, dismissed suggestions Israel was guilty of war crimes.
"Hamas bears sole responsibility for the violence," he told the security council.
Jacky Rowland, reporting from Gaza, said that Hamas officials were being targeted in the ongoing Israeli operations.
She said: "Police officers have been told to leave their posts, as there is mounting speculation that police stations are now targeted by the Israeli military."
"Also, Haniya appeared on Friday - the first time he was seen in public for quite a while, adding to the possibility that Hamas officials are avoiding public appearances."
Al Jazeera's John Terret, reporting from New York, said that members of the UN security council discussed a draft resolution regarding the conflict in Gaza.
He said: "The Libyan delegation representing the Palestinians have put forward a draft, and it is now subject to debate from the members."
But diplomats have reportedly said that the security council was unlikely to adopt a Libyan resolution that condemns Israel's killing of civilians, but makes no mention of Palestinian rocket fire.
Western diplomats say the Libyan resolution would not pass unless it was amended to condemn the Palestinian rocket attacks and dropped language suggesting Israel was guilty of terrorism.
The blooshed in Gaza jeopardises US-backed peace talks between Israel and the Palestinian Authority.