Obama was sworn in as US president two days after Israel and Hamas called separate unilateral ceasefires after a three-week Israeli offensive in the Palestinian Gaza Strip.
Prior to his inauguration, Obama was criticised for remaining silent on the conflict, in which more than 1,300 Gazans and 13 Israelis died.
Egypt and Jordan both have diplomatic relations with Israel and Abbas is considered to be Israel's and the West's partner in peace. Obama did not speak to the overthrown Hamas government, nor other Arab leaders who support the group's armed struggle against the Jewish state.
During the "warm" telephone calls, Obama "emphasised his determination to work to help consolidate the ceasefire by establishing an effective anti-smuggling regime to prevent Hamas from re-arming," and by working with the Palestinian leadership to help create a "a major reconstruction effort for Palestinians in Gaza," Gibbs said.
Marwan Bishara, Al Jazeera's senior political analyst, said that Obama highlighting the need to stop the smuggling of weapons into Gaza would be problematic.
"If this is the tone of voice of which Barack Obama is speaking to the leaders in the Middle East, it is going to be disappointing to a lot of people in Gaza.
"There is nothing about his understanding and sympathy to the suffering of those people who were under shelling and bombardment for more than three weeks, and hence no true approach that reflects a sincerity and change in the American approach to Israel-Palestine."
However, Bishara said that Obama's speedy contact with the Middle Eastern leaders is positive and showed a responsiveness and a responsibility for the events in the region.