Sergeant Nicolas Mazier, an air force paratrooper, "fell in combat while carrying out his mission", the Elysee Palace said in a statement, expressing Macron's "profound respect" for his "sacrifice".
It said other soldiers from the unit were also wounded.
France's chiefs of defence staff said in a statement that Mazier was mortally wounded and four more troops hurt during "a reconnaissance operation supporting Iraqi forces around 100 kilometres (60 miles) north of Baghdad".
When the Iraqis were attacked by "a group of terrorists in fortified positions", the French "immediately responded to support their partners, inflicting serious losses on the enemy," the military headquarters added.
An Iraqi security source in northeastern Kirkuk said that Iraqi and French troops were ambushed by Islamic State group jihadists late Monday in neighbouring Salah al-Din province.
The battle lasted "more than five hours" and saw three members of Iraq's anti-terrorist forces wounded as well as the French casualties, the source added.
Mazier is the third French soldier to lose his life this month in Iraq, where the country's troops form part of Operation Inherent Resolve, a years-long campaign by an international coalition against the Islamic State group.
Two men, Sergeant Baptiste Gauchot and officer Nicolas Latourte, were killed in separate incidents in August, the first in a traffic accident and the second during a training exercise.
Around 600 French troops based across the Levant and Gulf region are involved in Inherent Resolve.
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