Martinez, who withdrew from the Spain squad for their Nations League games due to a swollen right knee, played the full 90 minutes in Barcelona's 3-0 win on Thursday which moved the league leaders three points clear in La Liga.
"The club understands that the participation of Inigo Martinez in yesterday's match violated article 5 of Annex I of the FIFA Regulations on the Status and Transfer of Players," Osasuna said in a statement.
"(It) specifies that a player who does not join or leaves his national team for medical reasons may not play matches for his club during the five calendar days following the end of the international period.
"Osasuna believes that Inigo Martinez, whose absence from the Spanish national team was due to a medical leave, was not eligible to play in yesterday's match in accordance with FIFA regulations."
The result left Osasuna 14th in the standings, six points clear of the safety zone.
Barcelona were originally meant to play Osasuna on March 8th, but the death of Barca club doctor forced the game to be rescheduled to March 27th just four days after Spain's match as there was no other date available.
Both clubs had appeals for the fixture to be postponed rejected, with Barca unable to field some of their players such as one of their top scorers Raphinha, who had just returned from international duty.