"I hope the world will understand that the security council is not the place to get any arrangement, any concessions, any resolution. This has to be taking place on the ground," she said.
Foreign ministers from several Arab nations, the US, Britain and France were also attending Tuesday's meeting.
Ban Ki-moon, the UN secretary-general, said at the meeting's opening that the "only" answer to the crisis was "an end to the violence ... and a political way forward".
He also said he intended to visit Israel, the Palestinian territories and other nations in the Middle East later in the week for further meetings on the crisis.
Condoleezza Rice, the US secretary of state, said the US remained "deeply concerned" about the suffering of Palestinians and Israelis in the crisis and "understood" the urgency for an end to the fighting.
However, she said the council needed "to find a solution ... that does not allow Hamas to use Gaza as a launching pad for rockets".
Rice also called for the reopening of crossings into Gaza and the prevention of Hamas's rearmament, adding that she and George Bush, the US president, "remain dedicated" to the goals of Israel's security and a Palestinian state.
"The time has long since come when Palestinians who deserve to live in their own state should get that state ... and Israelis should have that peace and security," she said.
Pressure on Arab leaders
Al Jazeera's Ghida Fakhry, reporting from New York, said that it was thought unlikely that any vote would take place on the proposed resolution soon, as the current reported document was very complicated.
It could take some time for all parties involved to iron out their differences over the main points being urged in any resolution - a ceasefire, the lifting of the blockade of Gaza and a "monitoring mechanism" for any ceasefire, our correspondent added.
|Rice said any UN solution should prevent
Hamas firing more rockets into Israel [AFP]
But it was unlikely that Abbas would achieve his goal of getting the resolution passed as the US consistently vetoes any resolution critical of Israel, and this would weaken the Palestinian president politically, our correspondent explained.
A delay in getting a resolution would also increase the pressure on Arab nations, she said.
"There has been much dragging of feet, but the increasing number of deaths in Gaza has increased pressure on Arab governments' need to come up with some resolution as soon as possible," our correspondent said.
Last week, the US blocked a Libyan-backed proposal for the UN to call for an immediate ceasefire in Gaza.
Analysts say both Israel and the US consider the Palestinian group Hamas a "terrorist" organisation and Israel fears any council resolution might equate the legitimacy of Hamas with its own legitimacy as a member of the UN.
The US shares Israel's concerns, and would likely use its veto on such a resolution if it were brought to a vote.
The latest council meeting follows a flurry of diplomatic meetings on the crisis earlier on Tuesday and on Monday.