By Margarita Antidze

TBILISI (Reuters) - Georgian President Mikheil Saakashvili held out an olive branch to Prime Minister Bidzina Ivanishvili on Friday, proposing an end to months of friction that threatens stability in the former Soviet republic.

Saakashvili struck a conciliatory tone in remarks to thousands of supporters at their first big rally since his party was swept from power by Ivanishvili's opposition movement in a parliamentary election last October.

The defeat set up a tense situation in the Caucasus state, a conduit for pipelines pumping Caspian Sea energy westward and a platform for geopolitical rivalry between the United States and Russia, which fought a brief war with Georgia in 2008.

Saakashvili is barred by term limits from seeking re-election this autumn and the president's powers have already been reduced, but his United National Movement wants to remain relevant.

"I'd like to extend a hand of friendship to them," Saakashvili said of Ivanishvili's government, addressing more than 10,000 supporters in the centre of the capital Tbilisi.

"If we, your opposition, are optimistic, you should be optimistic too - and not angry," he said.

But he got a cool reception from Ivanishvili's camp.

"Saakashvili could not become a president of the whole country. He is just a leader of one political team - and this team, as well as he himself, are in the past," said Eka Beselia, a lawmaker from Ivanishvili's Georgian Dream coalition.

The rally was framed as a new start for Saakashvili's party and a show of support for the pro-Western course set by the president, who was first elected after leading the peaceful "Rose Revolution" protests in 2003 that swept out the old guard.

Saakashvili cultivated close relations with Washington, and his efforts to bring Georgia into NATO were one of the factors that led to the five-day war with Russia in 2008.

Ivanishvili, a billionaire who made his fortune in Russia, fought off suggestions from Saakashvili during the parliamentary election campaign that he was a Kremlin stooge.

Demonstrators echoed that theme on Friday, holding posters that read, "Any country but Russia - no way, but the West", "Bidzina, go home!" and "Georgia's choice is Europe."

Ivanishvili has pledged to make relations with the West a priority, following in Saakashvili's footsteps, but says he also wants to improve ties with Moscow.

Russia recently agreed to open its market to Georgian wine and mineral water, which were banned in 2006 as tensions increased, and Ivanishvili wants an investigation into the war to see whether Saakashvili was partly to blame.

(Writing by Margarita Antidze; Editing by Steve Gutterman and Mark Heinrich)