* Obama says Gaddafi rule over
* Urges loyalists lay down arms to end bloodshed
* Cautions Libya rebels not to seek violent reprisals (Adds details, background)
By Laura MacInnis
CHILMARK, Mass., Aug 22 (Reuters) - President Barack Obama urged Muammar Gaddafi on Monday to end the bloodshed in Libya as pockets of his loyalist forces engaged in fierce fighting against advancing rebels.
Reminding the United States that Gaddafi had "murdered scores of American citizens," Obama interrupted his vacation to herald Gaddafi's fall and urged him to limit the killing.
"Although it is clear Gaddafi's rule is over, he still has the opportunity to reduce bloodshed by explicitly relinquishing power to the people of Libya and calling for those forces that continue to fight to lay down their arms," Obama said.
While rebels hunted for Gaddafi in Tripoli, some forces loyal to the autocratic leader were resisting.
"This is not over yet," Obama warned in a statement from the farm where his family is vacationing on Martha's Vineyard, an island off the coast of Massachusetts.
Vowing the United States would be a "friend and partner" to help the emergence of a democratic Libya, Obama also cautioned rebels against exacting revenge for Gaddafi's brutal rule.
"True justice will not come from reprisals and violence. It will come from reconciliation and a Libya that allows its citizens to determine their own destiny," Obama said.
Analysts see risks that Islamic militants with links to Al Qaeda may take advantage of instability after the crumbling of Gaddafi's control over the country and gain a strong footing in the oil-producing nation.
Obama made plain that the United States would oppose any group within the loose coalition of rebels that has fought Gaddafi from imposing its power over other parts of Libyan society.
CALLS FOR "INCLUSIVE TRANSITION"
"Above all we will call for an inclusive transition that leads to a democratic Libya," Obama said.
Democrat Obama took heat from both his own party and Republicans for sending U.S. warplanes into harms way as part of a NATO campaign to bombard Gaddafi forces and shield Libyan civilians from harm.
But with the rebels hunting Gaddafi in the capital of Tripoli, Obama allowed himself some of the credit for Gaddafi's ouster.
"In the early days of this intervention the United States provided the bulk of the firepower, and then our friends and allies stepped forward," he said, referring to his order in March to send U.S. warplanes into battle over Libya.
Although he did not go into details about what help the United States would be prepared to offer Libya, Obama said a top priority would be humanitarian aid to the wounded.
He did spell out that U.S. engagement would continue to be part of a multinational effort and praised the role NATO had played in the campaign against Gaddafi.
"NATO has once again proven it is the most capable alliance in the world and its strength comes from both its firepower and the strength of out democratic ideals," Obama said.
He also reminded a U.S. public that has not paid much attention to the country's involvement in the NATO campaign against Gaddafi that the Libyan dictator had American blood on his hands, in an apparent reference to the 1988 Pan Am airline bombing over Lockerbie, Scotland.
"Gaddafi's regime has murdered scores of American citizens in acts of terror in the past. Today we remember the lives of those who were taken in those acts of terror," he said. (Reporting by Laura MacInnis and Alister Bull; Additional reporting by Patricia Zengerle in Washington; Editing by Peter Cooney and Christopher Wilson)
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