Regarding the uncertainty expressed in your second sentence, rest assured that using the notion of getting an "invitation to [join the Army/become a soldier/fight a war]" to euphemistically characterize "being drafted" is perfectly okay in English (at least in American English, where the invitation is often described, as it is here under the caption & in paragraph three, as having been issued by/received from "Uncle Sam."
If you are trying to at least approach this euphemistic tone in French, although "appelé" is used officially (i.e., non-euphemistically) to describe matters of conscription in French (just as "called-up" is in English), "appeler" can include the notion of "invitation" (which is included in your English version) and therefore, I would nevertheless suggest something like the following:
Mon grand-père a été appelé à la
(paraphrased from Six ans de bonheur, d'angoisse et d'espérance,
by Pierre Martin, via Googlebooks)