The loss has hurt Madrid's hopes of winning the Spanish league title, leaving the team five points behind leader FC Barcelona with eight games remaining.
"I'm going to report these calls to the police because I want them to stop," Turienzo Alvarez told Punto Radio. "They are strong threats, both against my life and that of my family members."
Turienzo Alvarez, who is forbidden by refereeing rules to discuss his decisions, said he feared more for his relatives than himself.
"I have a daughter of six who yesterday came home from the park in tears. My daughter said, 'The children say I have hurt Madrid.' My heart sank," the 40-year-old referee said.
Spain's Referees' Committee is reportedly considering asking the federation to take action against Mijatovic.