Armed mobs of government loyalists took to the streets of Ivory Coast's main city, Abidjan, in anger against France.
Ivory Coast lost two Su-25 ground attack planes in the French attack
Rioters attacked a French school and army base as Paris moved to disable the Ivory Coast air force following an air attack on French peacekeeping force.
Nine French soldiers and a US citizen were killed when Ivorian planes hit a rebel stronghold in the north.
The UN Security Council condemned the attack, and voiced support for French and UN forces in Ivory Coast.
Paris has dispatched an extra two companies of troops to beef up a force of 4,000 already deployed since the end of the civil war last year, and has also redeployed three jet fighters to the region.
French forces earlier destroyed two Ivorian bombers at an airbase in the capital, Yamoussoukro, along with two Russian-built Sukhoi 25 and three Mi-24 helicopters.
In a telephone call to President Laurent Gbagbo, France's Foreign Minister Michel Barnier said a political solution must be found, and stressed that "violence leads to nothing," a ministry statement said.
The BBC's world affairs correspondent, Mark Doyle, says this is the most serious crisis between France and its former colony since independence in 1960.
Ivory Coast was for many years a tolerant melting pot of religions and ethnic groups, but a coup in 1999 followed by civil war ended all of that with a vengeance, our correspondent says.
'French go home!'
Unconfirmed reports speak of explosions and heavy gunfire in Yamoussoukro on Saturday evening.
The earlier attack on its airport ignited anti-French feeling in Abidjan - a much bigger city - where mobs loyal to President Gbagbo went on the rampage.
Brandishing axes, machetes and clubs, they roamed the streets shouting "French go home!" and "Everybody get your Frenchman!" as French property was looted.
At Abidjan's airport, French and Ivorian forces exchanged shots and a French military plane was reportedly damaged.
Troops guarding a French base in the city fired tear gas at a crowd protesting at the destruction of the Ivorian planes and a French secondary school was also set alight
President Gbagbo has appealed through a spokesman for an end to attacks on French interests pending an investigation into Saturday's events.
A statement from the presidential palace in Paris said that two extra military companies were being sent to Ivory Coast to boost the French force there.
President Jacques Chirac ordered the "immediate destruction of Ivorian military aircraft used in recent days in violation of the ceasefire".
IVORY COAST'S PEACE UNRAVELS
29 Sept: Ivorian parliament fails to agree citizenship laws, which were a key requirement of the January 2003 peace deal
13 Oct: Ivorian rebels say they will not disarm, as planned, until immigration laws are changed
28 Oct: Vendors selling newspapers accused of supporting the opposition are attacked by pro-government militants in Abidjan and southern towns
The New Forces order eight rebel ministers to return to the rebel-held north, saying it had discovered the government smuggling arms across its territory
4 Nov: Government launches air strikes on rebel-held territory in north
5 Nov: More government air strikes and clashes on the ground in north, as unrest erupts in Abidjan
Nov 6: French forces destroy two government warplanes after an air strike leaves French soldiers dead
France also redeployed three Mirage jets, usually based in Chad, and a supply plane to Gabon, closer to Ivory Coast.
At least one Ivorian Sukhoi 25 bomber attacked a position of France's Unicorn peacekeeping force in the rebel stronghold of Bouake at 1300 local time (1300 GMT) on Saturday.
Eight French soldiers were killed immediately along with an American, believed to have been a missionary, while 23 soldiers were injured and evacuated to Abidjan. A ninth soldier later died of his wounds.
At about 1415, French forces launched an attack on the aircraft on the ground at Yamoussoukro.
Without giving details of the airport attack, the defence ministry said the army had "responded in a situation of legitimate defence" and was seeking "the immediate end of combat".
The African Union has urged both the government and rebels to refrain from any further violations of the truce they signed last year after a bitter civil war which split the country.