Abusive comments on the 29-year-old substitute's Instagram account poured in after Germany won 2-1 in the 95th minute in Sochi on Saturday.
“I can be criticised for my performance… but there is a line, and that line was crossed yesterday,” Durmaz said in a statement he read out to reporters at the team's Black Sea coast base in Gelendzhik on Sunday.
“When you threaten me, when you call me a “blatte” (a pejorative word for a dark-skinned foreigner), an 'Arab devil', a 'terrorist', 'Taliban', then you have gone far beyond the limit,” he said.
Durmaz, who was born in Sweden to Assyrian parents who emigrated from Turkey, said his family and children had also been threatened.
“Who the hell does such things? It is completely unacceptable,” he said.
“I am Swedish and am proud to play for Sweden. I will never let any racists destroy that pride. We must disregard all forms of racism”.
Sweden coach Janne Andersson said the squad “stand 100 percent” behind Durmaz.
Sweden face Mexico in Yekaterinburg on Wednesday, with the chance to progress to the knockout rounds.