Nour Odeh, Al Jazeera's correspondent in Gaza, said the ceasefire between Hamas and Fatah was generally holding. Sporadic gunfire was heard near the Hamas-controlled foreign ministry, but no injuries or wider unrest have been reported, she said.
Odeh said that the ceasefire instruction from the two factions would take time to trickle down to the fighters on the ground.
She added that the next few hours would see if gunmen or politicians would dictate the ceasefire agenda.
Against the backdrop of the ceasefire, Abbas explained his call for fresh elections.
He told the news conference: "On Saturday, I called for early presidential and legislative elections. With the current impasse I felt it was essential to allow the people to have their say on a platform that achieves Palestinian national interests.
"We want to examine the will of the people. Do they still trust those they have chosen?"
Abbas said he was still open to the formation of a unity government of technocrats.
The West has sought to bolster Abbas, who favours a two-state solution to end conflict with Israel.
Western sanctions, imposed because of Hamas' refusal to recognise Israel and renounce violence, have severely hobbled the Hamas-led government.
"We have been in a crisis for nine months. People cannot wait for long...People are suffering from the economic social and security situation"
Mahmoud Abbas, Palestinian president and leader of Fatah
An Israeli trade embargo has also severely limited the flow of funds through the occupied Palestinian territories.
"We have been in a crisis for nine months. People cannot wait for long," Abbas said. "People are suffering from the economic social and security situation."
Abbas called for increased international efforts to reach a permanent solution to the Palestinian-Israeli conflict, and said a meeting with Ehud Olmert, Israel's prime minister, was necessary.
"We have to meet. We need each other, and we have to deal with our problems," he said.
Reiterating his support for Abbas, Blair said the international community's job was to "support the people of moderation".
"If the international community really means what it says about supporting people who share the vision of a two-state solution, who are moderate, who are prepared to shoulder their responsibilities, then now is the time for the international community to respond," Blair said.
Fatah and Hamas accused each other of violating a Gaza ceasefire but said it was still in effect.
Officials from Hamas and Fatah were expected to meet on Monday to try to cement the ceasefire. Previous deals to end internal fighting this year have quickly collapsed.
Blair arrived in Israel just before Hamas and Fatah announced the deal on Sunday night. He will hold talks later with Olmert in Jerusalem.