His visit is part of a regional tour aimed at helping Israelis and Palestinians “end the violence and find a way forward”, the UN said.
Five Israeli soldiers and one Israeli civilian have died since Prime Minister Benjamin Netanyahu launched the military offensive on 8 July.
The Israel Defense Forces (IDF) sent ground troops into Gaza on Thursday after 10 days of heavy air and naval barrages failed to stop rocket fire from Gaza.
A UN official said the number of people fleeing the violence in Gaza was much higher than expected
Israel says the ground operation is necessary to target a Hamas tunnel network, which it says it could not do only from the air.
Lieutenant Colonel Peter Lerner, an IDF spokesman, said the ground offensive was being expanded “to restore security and stability to Israel’s residents and citizens”.
Residents in Gaza reported hearing explosions throughout the night on Saturday.
Meanwhile, the United Nations warned that it was running out of supplies to help more than 50,000 Palestinians who have sought shelter at its schools in Gaza.
A UN official said the number of people fleeing was much higher than expected, with both the Israeli and Egyptian borders closed to Gazans.
Fears over civilian deaths
Diplomatic efforts to secure a ceasefire involving, among others, Egypt, Qatar, France and the UN, have failed to make any progress.
French Foreign Minister Laurent Fabius met Mr Netanyahu on Saturday, but said that attempts to agree a ceasefire had failed.
“Sadly I can say that the call for a ceasefire has not been heard, and on the contrary, there’s a risk of more civilian casualties that worries us,” he told press.
Qatar is expected to host a meeting between President Abbas and Ban Ki-moon on Sunday before the UN chief continues on to Kuwait, Egypt, Israel, the Palestinian Territories and Jordan.
Mr Abbas is also due to meet Hamas leader Khaled Meshaal in a bid to convince the Islamist group to agree to Egyptian efforts to end the fighting.
Hamas rejected an Egypt-brokered ceasefire last week, saying any deal with Israel must include an end to a blockade of Gaza. (BBC)