Abbas spoke a day after a small group of leaders of the ruling Fatah movement chose him as its candidate for 9 January elections for Palestinian Authority president, despite demands by Fatah's young guard that a primary be held.
The speech marked the first time since Arafat's death on 11 November that Abbas outlined his views on the conflict with Israel.
Abbas appeared to be sending a message to Fatah's new guard that he would stand tough in future talks with Israel, despite his pragmatism and opposition to violence.
Abbas' ideas about a peace deal with Israel have always been close to those of Arafat: a Palestinian state in the West Bank and Gaza Strip, with East Jerusalem as a capital, and Israeli recognition of the right of return of roughly four million refugees and their descendants.
The refugee issue has contributed to the failure of previous peace talks. Israel has said it is willing to repatriate a few thousand refugees as a good-will gesture, but that absorbing millions would destroy the Jewish character of the state.
Promise to Arafat
At a memorial ceremony for Arafat at the Palestinian parliament, Abbas said he would walk in the footsteps of the late Palestinian leader. "We promise you [Arafat] that our hearts will not rest until we achieve the right of return for our people and end the tragic refugee issue," he said.
Fatah fighters, meanwhile, said on Tuesday they would challenge the nomination of Abbas as the presidential candidate.
Some Fatah fighters say they
will oppose Abbas' nomination
Some of the members of the young guard are pushing for Marwan al-Barghuthi, jailed by Israel, as the movement's candidate, arguing that he has a better chance of winning.
Al-Barghuthi's wife, Fadwa, said her husband would decide early next week whether to run as an independent.
Abbas, 69, represents the older group of politicians who returned with Arafat from exile in 1994, while al-Barghuthi, 45, leads the Fatah activists who grew up in the West Bank and Gaza Strip.
Arafat held three top jobs - PLO chief, leader of Fatah and president of the Palestinian Authority. Immediately after his death, Abbas was chosen as PLO chief. If he is also elected as Palestinian Authority president, he would be transformed from interim leader to Arafat's successor.
The decision to nominate Abbas was made on Monday evening by 13 members of Fatah's Central Committee, which is dominated by the old guard.
Role for Al-Barghuthi
Palestinian Cabinet minister Qadura Firas, an al-Barghuthi loyalist, said the Fatah candidate should be chosen in a much larger forum. However, other members of the young guard said they were ready to recognise Abbas.
Palestinian Foreign Minister Nabil Shaath urged al-Barghuthi not to run. "I think Marwan al-Barghuthi is a disciplined man.
Marwan al-Barghuthi is serving
five life terms in an Israeli jail
"He is committed to the decisions of Fatah," Shaath said. "Marwan will have an important role in the future."
Al-Barghuthi is serving five consecutive life terms for what an Israeli court said was his involvement in shooting attacks that killed four Israelis and a Greek monk. Al-Barghuthi, a former Fatah leader in the West Bank, has denied a role in the attacks.
Israel has said it would not release him.
Abbas served as Arafat's first prime minister in 2003. However, he resigned after just four months in power, frustrated with Israeli policy and Arafat's refusal to grant him real power.
The nomination of Abbas came just hours after outgoing US Secretary of State Colin Powell met Abbas and other Palestinian leaders in the West Bank town of Jericho and Israeli leaders in Jerusalem.
Powell was discussing arrangements for the election and signalling a new start in the post-Arafat era.
Colin Powell is pushing for the US-
backed road map peace plan
The United States and Israel both shunned Arafat, charging that he was involved in terrorism, and US involvement in Middle East diplomacy waned.
Now, Washington hopes for a moderate Palestinian leadership. Arafat's death is "a moment of opportunity that should not be lost," Powell said.
Shaath said Palestinian-US relations had improved. "We want to make use of it to proceed with the elections and proceed with a return to the peace process," Shaath said.
Powell said after meeting Israeli leaders that a successful election would allow work to resume on the US-backed road map peace plan, the stillborn blueprint of stages for an end to four years of violence and creation of a Palestinian state.