Shouldn't it be in nominative form, not (I assume) genitive?
не стало + gen. (literally, "it stopped being for smb. / smth.") means "this thing is not there anymore".
У меня нет машины // I don't have a car
Вчера у меня не стало машины // Since yesterday I don't have a car anymore
Вчера у меня не было машины // Yesterday I didn't have a car
Отца не стало в июне means "father is not there anymore since June", which is an euphemism for "father died in June".
Nominative is mostly used when noun acts as a subject. In your case "father" is an object, and there is no subject at all. Such sentences are called impersonal since there is no distinct subject.
In English "there is"/"it is" or similar constructions are used for this.
No you could ask why the accusative is not used then, and the reason is "не стало чего-либо/кого-либо" simply demands noun in genitive. One could also try to invoke the negation rule that explains change of accusative to genitive in negated sentences but "стать" is not a transitive verb.