The following attack deals spell damage, yes. It deals spell damage for just that one attack, though.
(Very late edit: Yes, one attack, not the player's entire turn; if you try to cast Adrenaline on Seraph, have him (it?) use Divine Shield, then cast Shard of Wisdom on him (must be done in that order), only the first attack deals spell damage; the next one deals physical damage.
You can try that yourself with an unupped Seraph. (An upped one will have 16 attack after one SoW, upon which Adrenaline won't have any effect))