Parliamentary elections will be held on 25 January, but officials visiting Rome said it was not clear if Israel would allow Palestinians to vote in Jerusalem, which is under Israeli control.
Abbas, at the end of a two-day visit to Italy, said on Saturday: "If we can't hold democratic elections in Jerusalem it will be difficult to do it in the other zones. At that point Israel will be responsible for the block of the democratic process in Palestine.
"The Oslo accords establish the right of Palestinians to take part in elections. The Palestinians in Jerusalem have to have the freedom to vote inside Jerusalem."
An official in the Israeli Foreign Ministry said in previous cases, Palestinian residents of East Jerusalem, annexed by Israel after the 1967 Middle East war, were allowed to vote "by proxy" at Israeli post offices - a procedure agreed by both parties.
Mark Regev, the spokesman, said: "Israel stands by the principles for the voting mechanism which were negotiated in the 1990s and hopes that the Palestinians will not try to unilaterally change them because this would not be a positive move."
Sharon not to co-operate if
Hamas contests elections
However, Israeli Prime Minister Ariel Sharon has said that Israel will not aid the ballot if the Palestinian Islamic resistance group Hamas stands for election. Abbas has refused Israeli calls to bar Hamas from the vote.
Abbas has also complained that Israeli military blocks in other areas will also make it difficult for Palestinians to vote.
He said: "The Israelis should remove all obstacles that hinder Palestinians from being able to take part in the vote freely."
Meanwhile, senior Fatah leaders have swept the group's internal polls or primaries amid charges of widespread fraud and irregularities.
According to results on Saturday, Jibril al-Rajub, a former security chief, won nearly 13,000 votes in Friday's elections in Hebron, which is the largest electoral district in the West Bank.
Rafiq Natsha, former Speaker of the Palestinian Legislative Council, Jamal Shubaki, the former minister of local governance and Nabil Amr, another former minister, were also elected.
Jibril al-Rajub secured 13,000
votes in Hebron
Fatah officials said elections in several areas, including Qalqilya and Salfit, were cancelled due to inadequate preparation and violent incidents.
Witnesses told Aljazeera.net that in Salfit, south-west of Nablus, armed men affiliated to the al-Aqsa Martyrs Brigades, Fatah's armed wing, fired in the air and burnt ballot boxes.
Last week, elections in Rafah and other places in the Gaza Strip were cancelled after armed men stormed voting stations.
The elections have been marked by allegations of vote rigging and fraud.
A Fatah activist in the town of Dura said the most common form of fraud was voting by people who were not members of Fatah.
The activist, on condition of anonymity, said: "Every candidate would bring his relatives, including elderly people. I saw a particular candidate bring his 90-year-old grandmother to vote. This is not really a genuine process."
Critics, including Hamas, accuse the Palestinian Authority of grossly abusing governance by holding elections in places where Fatah's chances are good while excluding areas where Hamas is powerful.
The PA denies the charge. It says the decision to have a multi-phase election is due to procedural, not political, factors.
Khalid Amayreh, Aljazeera.net correspondent in the West Bank, contributed to this report