The champions have demanded a "structural reform" of refereeing in Spain amid a withering assessment of the performances of match official Alejandro Muniz Ruiz and VAR Javier Iglesias Villanueva.
Real's grievances centre on Espanyol's Carlos Romero, who went on the score the game's only goal, after only being booked for what the club felt was a "brutal" challenge on Kylian Mbappe.
They were also unhappy that Vinicius Junior had a goal ruled out.
The letter, which is addressed to RFEF president Rafael Louzan, read: "The events that occurred in this match have exceeded any margin of human error or referee interpretation.
"What happened at the RCDE Stadium represents the culmination of a completely discredited refereeing system, in which the decisions against Real Madrid have reached a level of manipulation and adulteration of the competition that can no longer be ignored."
In a lengthy letter, Real claim Romero's tackle on Mbappe was "deserving of immediate expulsion" and the decision not to send him off, which was confirmed by VAR, was "manifestly erroneous". Real head coach Carlo Ancelotti described it as "inexplicable".
They also say the incident in which the Vinicius 'goal' was chalked off for a foul by Mbappe earlier in the move, should in fact have resulted in a penalty to them.
The letter continues: "The scandal generated by this match has once again had worldwide repercussions, with the international press denouncing the biased use of the VAR in Spain and the lack of credibility of Spanish refereeing."
Madrid currently top La Liga by a point from city rivals Atletico, with Barcelona four points behind.
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