The unidentified man climbed into the five-month-old animal's pen to pose for a photo on 10 December.
Nativity mule Platero died three days later after residents noticed it was struggling to stand.
Local animal rights activists in Lucena, Andalusia, said the donkey spent three days in "agony" and its injuries were like the "insides burst".
A spokesperson for PETA told The Huffington Post: "There is no excuse for the pain that this donkey endured from his internal injuries or for his horrible death, both caused because animals were seen as nothing more than living Christmas props.
"The selfish, reckless event organisers now have the blood of this poor donkey on their hands."
The police investigation follows complaints from the Asociación de Defensa del Borrico (Association for the Defence of Donkeys) and the Círculo Animalista de Podemos Lucena (Lucena Animal Circle).
Podemos Animalista Lucena released a statement on Facebook, saying Platero "was literally squashed by a man whose photo was spread by WhatsApp users calling it a great achievement".
Animal rights campaigners in Spain have called for the use of live animals in nativity scenes to be banned following the incident.