Up to 300 soldiers cross the border to clash with Kurdish separatist fighters.
Massoud Barzani, president of Iraq’s northern Kurdish region, refused to meet Rice because of the US position over Turkey sending soldiers into northern Iraq.
Barzani was set to meet Rice in Baghdad but decided to abandon the meeting over the Turkish incursion against bases held by the Kurdistan Worker’s Party (PKK), Nechirvan Barzani, prime minister of the Kurdish regional government, said.
“It is unacceptable that the United States, in charge of monitoring our airspace, authorised Turkey to bomb our villages,” he said.
Rice’s visit comes days after Arab and Kurdish parties in the ethnically and religiously diverse city won a deal under which Sunni Arabs ended their boycott of a provincial 41-member council.
However, ethnic Turkmen have continued to boycott the council.
Rice met representatives of the Kurdish, Sunni and Shia Arab, Christian and Turkmen communities during her visit.
On Monday, Barzani said his administration favoured delaying by six months a referendum on the future of Kirkuk, easing immediate tensions among the mixed population.
“The regional government is in favour of this extension,” Barzani said, after meeting in the central city of Najaf with Grand Ayatollah Ali al-Sistani, an influential Shia leader.
According to article 140 of the Iraqi constitution, the referendum had been due to be held by the end of 2007 to decide whether the region and its oil wealth should go under the control of the autonomous Kurdish government.
After visiting Kirkuk, Rice went to Baghdad to hold talks with senior Iraqi politicians.
The talks are thought to have examined action by the Shia-led government in encouraging Sunni Arabs to drop opposition to the US’ presence in Iraq.
Rice also held a meeting with Jalal Talabani, Iraq’s president, at his residence in the capital.