Both of Julian's parents were Christians, and he was a student of the Bible. He essentially rejected Christianity; that is the definition of "apostate". So the term is appropriate. If he had been a secular ruler, rather than politically targeting Christians in an effort to to reverse the religious course of the empire, it might not have been such a sticking point. A just ruler, perhaps-- but apostate.
By his own accord, Julian never accepted Christianity, so it could be said that the term apostate doesn't apply. Julian's father was not initially Christian, but a convert. He was the half-brother of Constantine who also famously converted. There is therefore no evidence of Julian ever being Christian.
This is of course mostly relevant if you consider the term apostate a pejorative, which is very much a Christian stance. Therefore only Christians at the time and later Christians make a case of arguing the point. It reminds me a bit of the nazi "accusing" Chaplin of being a Jew.