(Updates with call with Clinton and Saakashvili)
* Tycoon politician seeks to calm post-election tensions
* His supporters protest over local vote rigging claims
* Saakashvili supporters accuse them of intimidation
By Margarita Antidze
TBILISI, Oct 4 (Reuters) - The winner of Georgia's parliamentary election urged his supporters on Thursday to end street protests against alleged vote rigging by rivals allied to President Mikheil Saakashvili.
The billionaire leader of the Georgian Dream coalition sought to calm tensions after his supporters protested outside local election commission offices and were accused by their opponents of making threats to officials.
"I'd like to ask you to stop all street protests," Bidzina Ivanishvili said in comments to reporters.
"You should understand that we are not in opposition any more and we should move into buildings, courts and the parliament and clarify everything there."
Any post-election violence would damage Georgian Dream's image even before it takes power, especially as it has accused Saakashvili of heavy-handed tactics against the opposition and using force against protests during his nine-year rule.
Washington ally Saakashvili conceded his party lost Monday's poll before the final results were announced, raising hopes of a peaceful transition of power in the volatile Caucasus state.
Those hopes have been threatened by the protests by hundreds of Georgian Dream supporters angered by alleged voting irregularities in at least 10 districts outside the capital Tbilisi where candidates from Saakashvili's United National Movement (UNM) won.
Western governments have welcomed Saakashvili's acceptance that his party had lost an election which some had feared could ignite unrest in a country that is a conduit for Russian energy supplies to Europe.
But Saakashvili's UNM accused Georgian Dream supporters of trying to intimidate election officials in districts where the UNM won by a narrow margin. In some areas, Georgian Dream wants an election runoff between the top candidates.
"Such actions by the Georgian Dream represent an attack on the democratic process and on the right of the people to express their will," the UNM said in a statement.
It urged international observers to ensure Georgian Dream does not "discredit the freest and fairest elections held in Georgia in its history."
Ivanishvili, 56, plans to be prime minister, a post that would make him Georgia's most powerful official when reforms weakening the head of state take effect after a presidential election expected some time in 2013.
A political novice who made his business fortune mainly in Russia, Ivanishvili has acknowledged his six-party coalition is fragile and that he faces a difficult balancing act between the West and Russia, which welcomed his election triumph.
Ivanishvili says he wants to mend ties with Moscow and denies Saakashvili's accusations he is a Russian stooge.
He has repeatedly called for calm since the election, as has Saakashvili whose final term ends next year, setting up an awkward period of political cohabitation with Ivanishvili.
Saakashvili, a strong U.S. ally, sought NATO membership for Georgia and fought a brief war in 2008 with Russia, whose leaders portray him as unbalanced.
The Central Election Commission and local campaign groups called for Georgian Dream to refrain from violence and settle their complaints through the courts.
"Everyone has the right to appeal but this must be done in accordance with the law," European Union Ambassador Philip Dimitrov told reporters.
Georgia's electoral system allocates 77 of the 150 parliament seats according to party lists and the other 73 according to victories in individual constituencies.
With ballots from 99.95 percent of polling stations counted under the party list system, Georgian Dream had 54.88 percent and Saakashvili's United National Movement had 40.39 percent of the votes, the Central Election Commission said.
In individual races Georgian Dream won 39 seats against 34 seats for the UNM.
U .S. Secretary of State Hillary Clinton called Georgian President Mikheil Saakashvili and Georgian Dream coalition leader Bidzina Ivanishvili o n Thursday.
The State Department said Clinton discussed the parliamentary elections, praisi ng Saakashvili for conceding his party lost a nd co ngratulating Ivanishvili on his coalition's victory. (Reporting by Margarita Antidze; Editing by Robin Pomeroy)