Abbas sealed the pledge on Thursday in discussions with leaders of 14 Palestinian political factions including Hamas and Islamic Jihad.
The groups said they would still respond to any Israeli attacks.
Israeli warplanes broke the sound barrier over Gaza throughout the talks, which came a day after an Israeli aircraft fired three missiles at a Palestinian rocket crew in Gaza after mortar fire at a Jewish settlement.
That was the latest escalation of Israeli-Palestinian violence since Abbas coaxed resistance groups into the truce he agreed on with Israel in February. Israel wants Abbas to enforce calm to ease a planned Israeli pullout from Gaza starting in August.
"So far we are committed to calm ... but if they (the Israelis) violate it, we will respond. If they abide by it, we will abide by it," said Khaled al-Batsh, an Islamic Jihad leader.
Officials from Hamas said the group would maintain calm if Israel did the same. But a Hamas spokesman, Sami Abu Zuhri, said: "There will be a reaction to every (Israeli) assault."
In the latest flare-up, resistance groups vowed revenge after Israeli forces killed an Islamic Jihad leader and a Palestinian police officer in the West Bank on Tuesday. The army said the Islamic Jihad leader had planned to dispatch bombers.
Hamas official Khalel Nofal (3rd L)
attended the Gaza meeting
Rocket and mortar fire by resistance fighters later killed two Palestinian farm workers and a Chinese labourer at a settlement in Gaza.
Some Palestinian factions at the talks with Abbas on Thursday said his ruling Fatah party was not serious about political reforms.
Hamas is furious over Abbas's decision to postpone Palestinian legislative elections that had been scheduled for 17 July and in which it had been expected to mount a serious challenge to Fatah's dominance in parliament.
Hamas says it is committed to the truce, which has reduced violence. But its agreement hinges partly on Abbas's promise of internal reforms.
Hamas sees the election delay as a violation of that deal.
Abbas, under pressure to set a fresh date for the legislative ballot, agreed at the meeting to urge parliament in writing to approve an election law in line with understandings reached with factions in ceasefire talks in Cairo in March.
Opponents of the vote delay see it as an attempt by the old guard within Fatah, plagued by accusations of corruption, to gain time to shore up support.
The elections are not expected before Israel completes its evacuation of all 21 settlements in Gaza and four of 120 in the West Bank in a plan Israeli Prime Minister Ariel Sharon bills as "disengagement" from conflict with Palestinians.