I don't think virga is related to virile. In fact, virga is related to virgin.

virile: 1490, from O.Fr. viril, from L. virilis "of a man, manly," from vir "a man, a hero," from PIE *uiHro "freeman" (cf. Skt. virah, Avestan vira-, Lith. vyras, O.Ir. fer, Welsh gwr, Goth. wair, O.E. wer "man"). Virile member for "penis" is recorded from 1541. Virility "period of manhood" is attested from 1586; meaning "manly strength" is recorded from 1603.


virgin: c.1200, "unmarried or chaste woman noted for religious piety and having a position of reverence in the Church," from O.Fr. virgine, from L. virginem (nom. virgo) "maiden, unwedded girl or woman," also an adj., "fresh, unused," probably related to virga "young shoot." For sense evolution, cf. Gk. talis "a marriageable girl," cognate with L. talea "rod, stick, bar." Meaning "young woman in a state of inviolate chastity" is recorded from c.1310. Also applied since c.1330 to a chaste man. Meaning "naive or inexperienced person" is attested from 1953. The adj. is recorded from 1560 in the lit. sense; fig. sense of "pure, untainted" is attested from c.1300. Virginity is attested from c.1303, from O.Fr. virginite, from L. virginitatem (nom. virginitas), from virgo.
Distraught pretty girl: "I've lost my virginity!"
Benny Hill: "Do you still have the box it came in?" [sic]

So I think virga gets its rain sense from a comparison between an early stem or shoot from a plant, and the meager rainfall that evaporates before reaching the ground.