The Turkish team was kept waiting for several hours at passport control at Reykjavik airport on Sunday and subjected to extensive security checks, according to accounts from the players carried by Turkish media.
Presidential spokesman Ibrahim Kalin took to Twitter to show his anger, slamming the "disrespect" shown to the national team and saying the treatment of the Turkish players was "unacceptable".
Fahrettin Altun, communications director at the Turkish presidency, also said the treatment was "not in line with diplomatic courtesy or sportsmanlike conduct."
The foreign ministry issued a diplomatic protest note to Iceland via the Norwegian embassy. Icelandic officials confirmed to AFP that they had received the protest but did not provide further details.
Isavia, the company who manages the airport, said the extended checks had been made for security reasons, since the airport the players originally departed from, Konya, is not party to the same security rules as airports within the EU or countries that have a related agreement.
The company said it was "under obligation to carry out a security search of all passengers coming from such airports," regardless of nationality.
"As a rule, the security search does not take long. Last evening, however, the search took longer as a search had to be made for electronic devices and liquids in unusually many bags," the company said in a statement.
The Icelandic Football Federation's security team told the news website Visir that players from the Icelandic team had experienced similar checks when they returned from Turkey after a match two years ago.
The incident comes right after Turkey defeated world champions France 2-0 in Konya in a Euro 2020 qualifier on Saturday.