göra is not the future tense, it's the infinitive.
gör is the present. In Swedish, we often use the present tense to mean future. We could have said …annars kommer jag att göra det, but … annars gör jag det is more natural. You just have to guess from context if a Swedish verb in the present tense refers to the present or the future. Here it's pretty clear that we're talking about future, so the English translation uses will.
"Det" can be translated to "it" or "that," but "it" and "that" are not interchangeable in English.
It has to be "it" in this sentence in English, because "it" refers to the task of calling her. "Call her! Otherwise I will do that" indicates that the thing you will do is something else, not calling her.