How does Romeo show respect to women, as seen in Shakespeare's Romeo and Juliet?
It's actually questionable as to whether or not Romeo really does treat women with respect. There is a great deal of evidence suggesting that Romeo is really the sort who would prefer to jump in through a woman's bedroom window for a romantic tryst than do things the honorable way, which is marriage. Romeo is only respectful towards Juliet because she demands it.
We first see evidence of Romeo's interests in sexual liaisons with respect to his behavior towards Rosaline. First, we learn from Romeo's father that Romeo has been seen morning after morning, at dawn, under a grove of trees in a certain part of town, crying. He has also been staying up all night long and then going to bed in the morning. Ask yourself, why? Where is he? What is it he's been doing? He is most likely being seen in the same part of town in which Rosaline lives. The grove of trees is most likely a spot where he can stand and see her bedroom window. Plus, he is staying up all night long and crying because he is being sexually rejected by her. We see further evidence of his purely sexual interests when later he tells Benvolio that Rosaline:
... hath Dian's wit,
And, in strong proof of chastity well arm'd,
From Love's weak childish bow she lives unharmed. (I.i.211-13)
Diane is the virgin Roman goddess of hunting and childbirth. Hence, what Romeo is saying here is that Rosaline is behaving like Diane and vowing to remain a virgin. Romeo's reference to chastity with relation to his feelings of love makes it very evident that his feelings are really only of a sexual nature. Had Romeo been interested in honorable marriage rather than a sexual liaison, he would have gladly mentioned marriage rather than chastity. Therefore, his references to Rosaline make it evident that, being young, Romeo is really only interested in taking advantage of women rather than being respectful and honorable through marrying them.
However, Juliet changes things. While at first it's very clear that Romeo is also thinking of Juliet purely sexually and confusing love with sexuality, it is also very clear that Juliet will not accept such treatment. When Romeo asks her, "O, wilt thou leave me so unsatisfied?," Juliet very wisely retorts, "What satisfaction canst thou have to-night?" (II.ii.131-32). However, what's interesting is that Romeo still does not directly refer to marriage. He only asks for her "faithful vow for [his own]" (133). It is Juliet who brings up the subject of marriage, asking him, "If that thy bent of love be honourable, / Thy purpose marriage, send me word to-morrow" (148-49). Juliet's determination to direct their conversation and relationship towards honorableness and marriage shows us that Juliet is the sort of woman to demand respect from a man, even Romeo.
Hence, while it is questionable that Romeo, in general, treats women with respect, Juliet changes his common behavior by demanding his respect.