Speaking to Aljazeera from Ram Allah, Hamas leader in the West Bank, Shaikh Hasan Yusuf, said Abbas's decision would be contrary to what was agreed upon during the Cairo dialogue.
Earlier on Saturday, Abbas issued a decree postponing the 17 July elections to "allow time to resolve a dispute over proposed reforms to the voting law".
He said the new date would be announced later.
Abbas is also due to meet Hamas leaders on the issue.
Media reports suggest that Abbas has decided to delay the elections by several months, a move that would give him additional time to fend off a challenge by Hamas.
A senior Palestinian Authority official, speaking on condition of anonymity, said the election date would probably be pushed back to November.
Abu Zuhri (R): Abbas did not
consult Palestinian factions
Abbas has also been squabbling with legislators over a proposed election law. He wants all candidates to compete at a "national" level, while some Fatah legislators want at least half of the legislators to be chosen from districts.
Aljazeera's correspondent said another reason behind the delay was that Fatah would elect representatives at the group's conference in August.
Hamas has emerged as a formidable rival in local Palestinian elections in recent months, indicating a strong performance in the national vote.
Hamas's Yusuf said the postponement of the Palestinian legislative elections would run contrary to Cairo decision and also against the interests of Palestinian people.
Yusuf said while Hamas respected the presidential decree postponing the elections, there should be a consensus on such important decisions as it affected the entire Palestinian population.
"All Palestinians should agree on postponing the elections," he added.
Yusuf said Hamas would hold consultations with all Palestinian factions, who had earlier agreed on the 17 July election date, to decide on the future course in the light of the Palestine president’s decision.
"All Palestinians should agree on postponing the elections"
Shaikh Hasan Yusuf,
Hamas West Bank leader
Earlier, Hamas spokesman Sami Abu Zuhri had described Abbas's postponement decision as unilateral.
"Abbas did not consult Palestinian factions," he said in a statement.
"This decision shows that the Palestinian Authority is not serious about its commitments. It risks causing chaos and giving a negative impression about the Palestinian nation", the statement added.