The club claimed "the laws of the game were not applied" in the quarter-final first-leg at the Park des Princes, which ended in a 2-2 draw.
Their gripe surrounds an incident in the second half when Barcelona players Javier Mascherano and Jordi Alba were hurt in a collision in their own area. Stark did not stop the game.
Barcelona spokesman Toni Freixa told the club's website: "The club has sent an institutional letter to UEFA regarding a series of events that occurred on the field of play, for it is our objective understanding that the referee clearly failed to apply the laws of the game.
"When two players from the same team are lying on the ground, the referee has to stop the game, and that didn't happen.
"Not only that, but he made them leave the field, which goes against the rules. This is a serious enough thing to happen in a Champions League match for us to write to UEFA and make sure it never happens again."
He added: "We are fully aware that referees can make mistakes and we don't feel unfairly treated, but what calls our attention is that in a high level competition, a referee didn't know the rules.
"We are not judging the refereeing, but a basic principle. A judge has to know the law. We hope this never happens again. Barca do not complain about referees, not a bit or a lot. We are denouncing an objective error of not applying the rules."