Without going into detail over the incident, Erdogan said an image of Ataturk and his own name were used "and these were the targets".
The message was written by a Norwegian national who was hired to work on staff during exercise, he said, adding that the individual was removed from the exercise as a outcome of the incident.
In a statement issued on Friday, NATO's Jens Stoltenberg said: "I apologise for the offence that has been caused".
The 40 Turkish soldiers participating in a North Atlantic Treaty Organisation drill in Norway were withdrawn after authorities were notified that Turkey's founder and first president, Mustafa Kemal Atatürk, and current President Recep Tayyip Erdoğan were both depicted as "enemies".
Erdogan said an "enemy poster", featuring his name on one side and a picture of modern Turkey's founder Ataturk, on the other, was unfurled at the training exercise in Norway, prompting a decision by Turkey's military chief and European Union minister to pull the troops out.
Turkey, which became a member of North Atlantic Treaty Organisation in 1952, is a key member of the alliance but tensions have grown in recent months over its crackdown after the failed coup and Ankara's increasingly close alliance with Russian Federation.
"There was an incident in Norway", Erdogan told ruling Justice and Development (AK) Party members in capital Ankara.More news: You could live longer if you had a dog
"They said they had chose to pull our troops out and will do so, so we told them to not stop and go ahead and take our 40 soldiers out of there".
"The individual in question was immediately removed from the exercise by the Joint Warfare Center, and an investigation is underway".
The Turkish leader said of the North Atlantic Treaty Organisation alliance: "There can be no such pact, no such alliance", the New York Times reported.
The country has the second largest military force in North Atlantic Treaty Organisation, and is involved both in coalition efforts against so-called Islamic State and in Nato's Afghanistan mission.
"The Norwegian national was removed from the exercise by JWC as a outcome of the incident".
"Turkey is a valued NATO Ally, which makes important contributions to Allied security".