Barca were unhappy with Stark's handling of the game and claimed the official 'did not know the rules of the game' after he allowed play to continue when Javier Mascherano and Jordi Alba collided and were both lying prone on the ground in the penalty area.
Regulations state that play should stop in that instance, while the official also allowed Zlatan Ibrahimovic's first goal to stand when it appeared he was offside.
"The club has written to UEFA due to a series of incidents on the pitch in the understanding that objectively speaking it was evident that the referee did not apply the rules of the game," Barca spokesman Toni Freixa told a news conference after a meeting of the club's board on Thursday.
"When two players from the same team are on the floor the referee should stop the match, which did not happen.
"What's more he then obliged them to leave the pitch, in contravention of the rules.
"An incident that is sufficiently serious in a competition like the Champions League to bring to UEFA's attention so that it does not happen again."
Freixa added: "We are fully conscious of the possibility of errors in refereeing and we do not consider ourselves badly treated," Freixa said.
"But it surprises us that in a high-level competition a referee does not know the rules.
"We are not judging the work of referees but the basic principle that a judge should know the law. We hope it does not happen again."