Misogyny is generally understood as a hatred of women, but feminist theory interprets misogyny as being shown through "discrimination, denigration of women, violence against women, and sexual objectification of women" (Dr. Mel Konner, "The Tangled Wing"). The essential point is that it should not be seen as simply hateful feelings felt towards women or as a general mistreatment of women. Dr. Mel Konner points out that women have been subjected to many horrible things, such as Hindu women being burned alive on their husband's funeral pyres, not because the men subjecting them to these tortures hate the women, but because the men claim to be protecting the women. Yet, the horrible treatments fall under the category of misogyny because they are a means of suppressing women. Duke Orsino certainly does show that his main interest in wanting to marry Olivia is to satisfy his sexual longings, which is sexually objectifying Olivia. In this sense, he does think through misogynous ideas. However, unlike misogynists, he does not treat Olivia as inferior. Instead, just like the poet Petrarch, which Shakespeare satirizes in the play, Orsino places Olivia on a pedestal and worships her; therefore, in that sense, he is not a misogynist. Furthermore, he does not continue to think as a misogynist by the time he falls in love with Viola, but rather values her not for her sexuality but for the honor and friendship she has shown him.
We especially see Orsino's sexual objectification of Olivia in the very first scene. We clearly see him referring to his sexual desires in this scene, especially when he likens his desires to hunting dogs pursuing his own heart, which he likens to a dear. He states that the moment he first saw Olivia, his heart became a "hart," meaning stag, being chased and tormented to be torn apart by his own desires, as we see in his lines:
O, when mine eyes did see Olivia first,
Methought she purg'd the air of pestilence!
That instant was I turn'd into a hart;
And my desires, like fell and cruel hounds,
E'er since pursue me. (I.i.20-24)
However, despite seeing her as a sexual object, this passage also shows just how much he has placed Olivia on a pedestal, which does not fit the definition of misogyny. Likening her beauty to something so purifying that it can heal plagues is certainly a means of putting her on a pedestal.
Furthermore, by the time he learns Viola is truly a woman, he no longer associates love purely with sexual desire nor does he still think with respect to the Petrarchean ideal of worshiping women; instead, he has learned a much more moderate and practical view of love. He has learned to love Viola for the loyalty she has shown him and the services she has done for him, such as courting Olivia for him even though she is in love with him herself. Due to Viola's loyalty, he now views women as much stronger than he did before, even viewing them as being on an equal footing as himself, as we see in his lines, "Your master quits you; and, for your service done him, ... you shall from this time be / Your master's mistress" (V.i.229-34).
Hence, we see that while Orsino may have had misogynous views at first, he develops as a character to accept new views of women.