Foreign Minister Avigdor Lieberman's party wants to ban Israeli Arabs from marking the anniversary of what they term "the Catastrophe" in 1948 when Israel was created as thousands of Palestinians marked the 61st anniversary of the Naqba that sparked an exodus of hundreds of thousands of refugees.
The ultranationalist Yisrael Beitenu party said it would propose legislation next week for a ban on the practice and a jail term of up to three years for violators.
"The draft law is intended to strengthen unity in the state of Israel and to ban marking Independence Day as a day of mourning," said party spokesman Tal Nahum.
The initiative could fuel racial tensions stoked by Lieberman's February election campaign call to make voting or the holding of public office in Israel contingent on pledging loyalty to the Jewish state.
Arabs, who make up 20 percent of Israel's population, said the allegiance demand was aimed at them and accused Lieberman of racism.
Israel celebrated its Independence Day this year on April 29, in accordance with the Hebrew lunar calendar. Palestinian refugees around the world and Israel's Arab citizens mark the Nakba on May 15, the day after the British mandate over Palestine ended in 1948.
Ceremonies in the occupied West Bank were held a day early this year because May 15 falls on Friday, the Muslim day to go to the Mosque and pray.
Holding Palestinian flags and large wooden keys to symbolize those of homes from which they were forced out of by Israeli forces six decades ago, demonstrators marched in the center of Ramallah in the Israeli-occupied West Bank.
"The right of return is sacred", "No peace without the right of return", proclaimed banners held by the marchers.
The demonstration was headed by political figures and religious leaders and began at the tomb of legendary Palestinian leader Yasser Arafat at the headquarters of the Palestinian Authority, today run by his successor, president Mahmud Abbas.
"I came here to show that we believe that one day we will return. If not me, then my son," said Mohammad Hassan, 79.
Abbas, on a visit to Syria, was to make a televised address later in the day to mark the Nakba.
The right-leaning government of Israeli Prime Minister Benjamin Netanyahu, in which Lieberman's party is a key ally, has not endorsed the Western and Arab-backed goal of Palestinian statehood.
It also firmly opposes the division of Jerusalem and the right of return of Palestinian refugees, and Netanyahu recently introduced a demand that Palestinians, as part of any future peace agreement, recognize Israel as a "Jewish state."
In the northern West Bank town of Nablus, about 2,000 people participated in a march, holding Palestinian flags tied with black ribbons as a sign of mourning.
In the Gaza Strip, its Hamas rulers prevented the Palestine Liberation Organization (PLO) from holding a ceremony to mark Naqba in the tiny territory that today is home to one million refugees.
A PLO committee said in a statement that Hamas police had prevented the group from holding any events to mark the Naqba, slamming the decision as "a violation of political and democratic rights."
Witnesses said Hamas police were deployed in force in the center of Gaza City, preventing any public gathering. No comment was immediately available from the Islamist group.
This year's Naqba fell during the first official visit to the region by Pope Benedict XVI, who during a visit to the West Bank on Thursday alluded to the commemoration and expressed deep sympathy with refugees.
"In these days ... (the longing for peace) takes on a particular poignancy as you recall the events of May 1948 and the years of conflict, as yet unresolved, that followed from those events," he said at Aida refugee camp outside Bethlehem.
"With anguish, I have witnessed the situation of refugees who, like the Holy Family, have had to flee their homes," he said.