The Spanish club has alleged that the official did not "apply the rules of the game" after he allowed play to continue when Javier Mascherano and Jordi Alba collided with one another and went to ground.
Stark has also been criticised for allowing a Zlatan Ibrahimovic goal to stand despite the striker appearing to be 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," said Barcelona spokesman Toni Freixa.
"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 went on to say that the club does not feel hard done by following Stark's alleged errors, but felt the matter should be investigated due to the high-profile nature of the competition.
"We are fully conscious of the possibility of errors in refereeing and we do not consider ourselves badly treated," he added.
"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."
The quarter-final first leg ended in a 2-2 draw.