The 54-year-old with a salt-and-pepper beard takes charge of Hamas as it seeks to ease its global isolation while not marginalising hardliners within the movement.
Barghouti is a popular figure in Palestinian society and is seen as a possible successor to Abbas.
Mishaal may have wanted not to expose so soon all Hamas' cards in the eventuality of a negotiation process that would lead to the establishment of a Palestinian state in the West Bank with East Jerusalem as its capital.
The PLO is considered the worldwide representative of Palestinians both in the Palestinian territories and the diaspora.
PM Ntanyahu said that the Hamas document showed that the Palestinian group remains unwilling to make peace with Israel. The six-page outline supports the creation of the Palestinian state in the West Bank and Gaza Strip within the 1967 borders. It also might deter Israel from attempting to assassinate the Hamas leader. This new document claims it now recognizes the border lines of Israel before the 1967 armistice, and that it declares independence from the Muslim Brotherhood.
Leila Seurat, a researcher at the Paris-based Centre for International Studies and Research, said the election of a Gaza-based leader signaled a shift for the movement.
Senior Hamas leader Ismail Haniyeh gestures before delivering a farewell speech for his former position as a Hamas government Prime Minister, in Gaza City June 2, 2014.
Netanyahu said that moving away from calling for "the genocide of all Jews to calling just for the annihilation of Israel - is that progress or moderation?"
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The White House also congratulated Moon, saying it looked forward to working with him to strengthen their longstanding alliance. Moon has said the decision had been made hastily and his government should have the final say on whether to deploy it.
In Washington last week, Abbas met Trump in their first face-to-face encounter, with the USA president saying he wanted to achieve the decades-long goal of Israeli-Palestinian peace.
A 42-article long document tells that Hamas is, above all, a Palestinian national group.
Later, Khaled Meshaal announced that Haniya was elected as the Hamas's new head in the internal Shura council elections.
"When the political leader of Hamas is from inside Gaza he will have more knowledge about Gaza's peoples' situation", commented Ahmad Okasha, a Gaza resident.
Hamas needs to change it actions, not only its rhetoric, if it wants to achieve political rehabilitation, but most importantly it must become part of the mainstream political structure to be recognized as a relevant player.
"Strengthening relations between Hamas and both Egypt and Iran will also be among priorities", he said.
But as the European official pointed out, he would have to "unite the movement behind him", in particular hardliners concerned with the recent moves. "This is the real Hamas", said David Keyes, a spokesman for Israeli Prime Minister Benjamin Netanyahu.
Hardliners yield influence in the strip, which has been under an Israeli blockade for 10 years.