Translation:Next time, he is not going to call you.
Not sure why this could not be used: "Next time, you are not going to call yourself"
Va can refer to the person being spoken to, but only if you're using the formal register (i.e. calling them by "usted") and if you were doing that you would not use "te" as that is the informal register (i.e. calling them by "tú"). Like amoussa says it would be "te vas" in informal and "se va" in formal.
I see the error I made. Would "se va a llamar" imply to call one's self, then?
I agree with lago. If that were the case, you would probably say, "Usted se va a llamar." There is a lot of potential confusion here, so if the meaning is not clear, the speaker will usually say, él, élla, or Usted, to make the meaning clear. For more emphasis, you would even say, "Usted se va a llamar a si mismo."
Would a structure like this not remove the He/She/It ambiguity in this and many other situations? La próxima vez que no llamados. The next time, you will not be called.
If anything surely this should be "The next time SHE is not going to call you" as "la" is female article? Although I agree "it" would also be appropriate. Just not "he".