Mr Ouattara, previously Ivory Coast's most prominent opposition figure, was among those candiates excluded from the election, after the supreme court ruled that he did not meet the nationality requirements.
Ivory Coast's electoral commission earlier declared that Mr Gbagbo had defeated General Guei in Sunday's presidential election.
But the country remains highly unstable, owing to the exclusion of several key political figures from the election.
Supporters of Mr Gbagbo's Ivorian Popular Front (FPI) - backed by members of the security forces - clashed with supporters of Alassane Dramane Ouattara's Rally of Republicans (RDR) party.
Mr Ouattara, previously Ivory Coast's most prominent opposition figure, was among those candiates excluded from the election, after the supreme court ruled that he did not meet the nationality requirements.
Mr Gbagbo secured the presidency after a day of unrest on Wednesday, when his supporters staged an uprising against General Guei, who had sent soldiers to halt the counting of votes.
Unity government
Speaking after his inauguration, Mr Gbagbo dismissed calls by the opposition and by international leaders for a more inclusive presidential election.
He promised, however to establish "a broad-based government of national unity".
This followed a joint television broadcast in which spokesmen for the FPI and RDR appealed for calm.
RDR spokesman Amadou Gon Coulibaly said the two party leaders would meet soon "to smooth out the difficulties".
Earlier on Thursday, paramilitary police joined Mr Gbagbo's supporters in an attack on Mr Ouattara's house.
An exchange of fire with Mr Ouattara's security men followed, in which a child is reported to have been killed.
Violence
Mr Ouattara fled and took refuge in the nearby residence of the German ambassador.
Presumed supporters of Mr Ouattara were being stripped naked and beaten by supporters of Mr Gbagbo.
Rival protesters set up burning roadblocks and smashed cars.
Police fired tear gas and stun grenades to disperse the crowds.
Mr Ouattara's spokesman Aly Coulibaly accused security forces of complicity in the violence.
"We're heading straight for a Rwanda," he said - a reference to the 1994 genocide backed by the then Rwandan Government.
Religious dimension
In other parts of the city, the fighting took on a religious dimension.
Mr Ouattara enjoys large support in the Muslim community and several mosques have been attacked. There are also reports of attacks on churches.
The recent developments are the latest in a year-long political crisis which has undermined confidence in a country once seen as West Africa's most stable and prosperous nation.
In total, more than 50 people have died in political violence since Sunday's election.
Army volte-face
BBC correspondent Barnaby Phillips says the army - which before the election was supporting General Guei - has quickly switched its allegiance to Mr Gbagbo.
But Ivory Coast's army has a history of in-fighting, and the situation is still far from stable.
The army has extended the state of emergency and the curfew remains in force until Satuday.
As battles raged in the suburbs, central Abidjan was almost deserted as people stayed away from work.
Clashes are also reported in several other cities in the south and centre of Ivory Coast.
General Guei's whereabouts are still unknown, though his family has fled to the nearby West African state of Benin.
Election call
UN Secretary-General Kofi Annan has called for a new election.
His statement echoed earlier calls by the Organisation of African Unity, and by la Francophonie - the group of French-speaking nations.
Mr Ouattara said fresh polls were essential as he believed only about 5% of the country had voted on Sunday as a protest against the military regime.