Saying he wants to see Israel and a Palestinian nation living side by side in peace, President Bush on Thursday praised Palestinian Authority President Mahmoud Abbas (search) and propped him up as the individual who can organize elections and help bring democracy to his people.

"President Abbas is a man devoted to peace and to his peoples' aspirations for a state of their own. And today the Palestinian people are closer to realizing those aspirations," Bush said in a Rose Garden news conference after a one-hour meeting between the leaders in the Oval Office.

"The way forward must continue to include democratic elections," Bush said.

Without mentioning the terror group Hamas (search), which is seeking political legitimacy by participating in Palestinian legislative elections in January, Bush also said the way forward includes rebuilding the Palestinian economy and confronting the threat of armed gangs.

While Bush credited Israel with its withdrawal from Gaza — a decision made unilaterally and completed ahead of schedule — the U.S. president urged the Jewish state to stop expanding settlements in the West Bank (search) and help Palestinians build their own economy.

He added that Israel shouldn't do anything to undermine its obligations to the "road map" created by the Quartet — the United States, Russia, the European Union and the United Nations — which aims to develop two states next to each other in the small swatch of land between the Mediterranean Sea and the Jordan.

"This means that Israel must remove unauthorized posts and stop settlement expansion. It also means that the barrier now being built to protect Israelis from terrorist attacks must be a security barrier rather than a political barrier," Bush said, adding that Israeli leaders must consider the impact of the barrier, built to protect Israelis from Palestinian terror attacks, on Palestinians who aren't terrorists or terror sympathizers.

In response, Abbas, who is popularly known as Abu Mazen, said the Palestinian Authority used great "energy" to make sure that the Gaza withdrawal wasn't interrupted by Palestinian agitators, insisted that Israel lift curbs on Palestinian travel in the West Bank.

The Palestinian leader also criticized Israel's security wall, particularly its location in Jerusalem, which the Palestinians want to establish as their capital. Abbas demanded that roadblocks be removed from the West Bank, Palestinian prisoners be released from Israeli prisons and Israeli forces withdraw to positions prior to Sept. 28, 2000, when the second intifada, or armed uprising, began against Israel.

In turn, Abbas said the Palestinian Authority is continuing to "ensure the calm and maintain it" and is increasing security, imposing the rule of law and public order, banning armed demonstrations and reinforcing reforms in the judiciary branch and administration.

"Mr. President, we reaffirm, again here today, our commitment to peace and negotiated stinian-Israeli conflict. The time has come that the Palestinian people will attain their freedom and independence. The time has come to move quickly toward the resumption of permanent status negotiations," he said.

With just 100 days until Palestinian legislative elections, Abbas said the January vote would be a landmark election that would demonstrate a "renewal process and rebuilding process" of the Palestinian political system.

"These elections will consolidate and reinforce the slogan I ran on during my presidential election which emphasized clearly the one authority, the one law, the one legal, legitimate law and political pluralism," Abbas said.

Even with Israel out of Gaza, the 1.4 million Palestinians who live there have little to no income and few job opportunities. While making demands on Israel, the Palestinian leader also praised international contributions to help build Palestinians' health, education and agriculture industries.

Abbas is also trying to find a way to raise travel curbs so Palestinians can go to election posts in January. Concerned about possible disruptions by Israel as it protects itself from would-be terrorists during the vote, the United States has asked Abbas to make sure candidates participating all renounce violence as a means of dealing with Israel.

In doing so, Hamas would be limited in its participation, which would also help Abbas gain support that would otherwise go to candidates from the terror group. Abbas, however, has consented to Hamas' participation in the election. He said all groups have accepted the "hudna" or calming, and all will become a part of the Palestinians political fabric.

Abbas is also meeting with House and Senate leaders on Thursday as well as Vice President Dick Cheney. Rep. Jane Harman, D-Calif., ranking minority on the House Intelligence Committee, said she hopes Abbas can contain Hamas and demonstrate strength.

"The goal is, is to make him stronger and help him succeed in forcing terrorist groups who are in Palestine to disarm. ... My hope is he will be stronger and have a big victory in the January election and he will be strong enough to operate this new state of Palestine by the rule of law," Harman said.

"Prime Minister Sharon is courageous and weathered the storms in Israel. Hopefully, Mahmoud Abbas will do the same in Palestine," she added.

Abbas also had wanted U.S. pressure to speed up the peace process, but Bush seemed to reject that.

"I'd like to see two states. And if it happens before I get out of office, I'll be there to witness the ceremony. And if doesn't, we will work hard to lay that foundation so that the process becomes irreversible," Bush said.

However, Bush said he was going to appoint a new coordinator to pursue an "enhanced mission" to help the Palestinians "end terror attacks, dismantle terrorist infrastructure, maintain law and order and one day provide security for their own state."

FOXNews.com's Sharon Kehnemui Liss contributed to this report.