"They are trying to deliver a fatal blow to the very existence of this country," Hollande said of the Islamist groups that control the north of Mali. "France, like its African partners, cannot accept this," he told a New Year gathering in Paris of French diplomats.
Meanwhile there are unconfirmed reports in Mali on Friday of the arrival of French military help.
Witnesses told journalists that foreign troops and weapons had already begun arriving by transport plane on Thursday to bolster government forces in central Mali, but it was unclear what country they came from.
France has so far offered only logistical support to the regional force.
The UN Security Council called for foreign troops to be sent quickly to Mali on Friday, to contain a new offensive by radical Islamists who control the north of the country and are vowing to capture more terrain.
The fresh fighting in Mali has re-ignited Western fears that the Al-Qaeda-linked militants could capture even more territory and turn it into the same type of sanctuary that Afghanistan was under the Taliban.
Diplomats at the UN said about 1,200 Islamist fighters were now 20 kilometres from the key central town of Mopti, the gateway between rebel-held and government-held areas.
The deployment of the African force to capture the territory back from the Islamists has been delayed by political tensions in Bamako, where the coup leaders still hold considerable influence.
There are also doubts about the how far the Malian army is ready and willing to defend the country.