The secretary-general also said he is "convinced and alarmed that this escalation will inevitably increase the already heavy suffering of the affected civilian populations."
Today, he added in a new statement, "Given the crucial juncture at which we have arrived in the search for a cease-fire, I appeal to all members of the international community to display the unity and commitment required to bring this escalating crisis to an end. "
Worldwide protests also have demanded an end to Israel's offensive.
In the West Bank today, an Israeli solider shot and killed a 22-year-old Palestinian man violently protesting by throwing rocks.
In Ramallah in the West Bank, a couple thousand Palestinians took to the streets in protest of the Israeli ground operation.
In Istanbul, Turkey, and Rabat, Morocco, enormous peaceful protests brought hundreds of thousands demonstrators onto the streets.
In London, where on Saturday thousands of anti-war protesters hurled shoes at the British Parliament building and at the entrance to Number 10 Downing Street, the prime minister's official residence, British Prime Minster Gordon Brown called the Israeli ground invasion "an escalation" and renewed his call for an immediate cease-fire in a BBC Radio interview.
"This is a very dangerous moment" said Brown. "This is a moment where all the hopes of the peace process are falling apart in the action that's being taken. So what we need is an immediate cease-fire."
Israel says the specific impetus for the offensive is to remove Hamas' ability to fire rockets and mortars at Israeli civilian targets. Since Israeli troops moved in, Hamas has fired 24 rockets and mortars into southern Israeli cities. One rocket injured a woman in Sderot after it scored a direct hit on her home.
Since last Saturday, 500 rockets and mortars have killed four Israelis and injured at least 73 others. Since August 2005, when Israel relinquished control of Gaza, the Israeli military says 6,500 rockets and mortars have been fired into southern Israel.
Despite Israel's ground onslaught, Hamas continued to fire rockets back at Israel -- more than 40 rockets today -- with one destroying a house in the town of Sderot and one giving an American delegation from New York City a brief scare.
New York Mayor Michael Bloomberg, Rep. Gary Ackerman, D-N.Y., and New York Police Commissioner Ray Kelly were in southern Israel on an official visit when rocket alarms sounded twice. None of them was injured.
Israel also is keeping its eye on its northern border.
"While we are fighting in Gaza, we keep an open eye on the sensitive situation on our northern border," Defense Minister Ehud Barak said in a televised speech Saturday. "We hope the situation will remain calm. Nevertheless, we are ready and alert to face any unwarranted development in that area."
Barak was addressing a possible offensive by Hezbollah as Israel conducts its ground invasion against Hamas. He is trying to head off what happened in 2006, when Israeli ground forces entered Gaza in an effort to free captured Israeli soldier Gilad Shalit.
Two weeks after Israel moved into Gaza, Hezbollah raided Israel's northern border with Lebanon, killing four soldiers and taking two others hostage. What followed was a major 33-day Israeli offensive into Lebanon which most Israelis feel was lost, despite their inflicting heavy casualties and destruction.