In the story itself, Giovanni never truly does reflect on himself and his own actions and responsibilities. Once he realizes that his nature has become imbued with the same poisons as Beatrice, he storms on, in a rage, wounding her irrevocably with his accusations. Once over, the narrator says, "Giovanni's passion had exhausted itself in its outburst from his lips." Giovanni feels too great a passion, and it prevents him from truly considering Beatrice's goodness and her loving nature. Then, he becomes so focused on "redeem[ing]" her, altering her nature so that she can be a part of the world, that he fails to consider how his own pride could actually damage her.
I'd like to believe that Giovanni would reflect on himself and realize his role in Beatrice's death, but nothing in the story leads me to believe that this might actually happen. I'm sure Giovanni will blame Rappaccini for making Beatrice poisonous in the first place, or Baglioni for offering him the potion which was to alter Beatrice's nature, because he seems unable to accept responsibility for anything.