There are about 56,000 Nato troops in Afghanistan battling the Taliban and al-Qaeda fighters.
The publication of the cartoons led to violent protests across the Muslim world and disputes concerning the Western world's upholding of freedom of speech and that freedom's ability to cause offence.
In Afghanistan some people died during riots against the publication of the set of drawings.
Islam generally prohibits any depiction of the prophet for fear that it may lead to idolatry.
Rasmussen defended the right to free speech throughout the events and did not apologise to Muslim nations.
Turkey, Nato's only majority Muslim member-nation, had opposed Rasmussen's appointment due to fears of heightened hostilities, but dropped their opposition earlier this month.
After his appointment, Rasmussen promised to consider religious sensibilities and called for a balance between free speech and religious respect.
"I was deeply distressed that the cartoons were seen by many Muslims as an attempt by Denmark to mark and insult or behave disrespectfully towards Islam or the Prophet Muhammad. Nothing could be further from my mind," he said.