Romeo's feelings of romantic love do not just extend to Juliet, but to his previous obsession, Rosaline. At the beginning of the play, Romeo is severely depressed because Rosaline does not show him the attention he wants. He tells his friend Benvolio:
[…] What sadness lengthens Romeo's hours?
Not having that, which, having, makes them short.
Out of her favour, where I am in love.
Romeo is obsessed with winning Rosaline's love, until Juliet catches his attention. His idea of romantic love is easily transferred.
During the balcony scene, Romeo climbs the wall and watches Juliet on her balcony. When he reveals himself to her, she asks him how he was able to get in without being caught. He responds:
Romeo: With love's light wings did I o'er-perch these walls;
For stony limits cannot hold love out,
And what love can do that dares love attempt;
Therefore thy kinsmen are no let to me.
In other words, Romeo's love for Juliet "gave him wings" and he was able to work up enough courage to sneak into her presence. Without the love he felt for her, he would not have been brave enough otherwise.
Before she finds out that Romeo has been watching her, Juliet speaks to herself on her balcony. She thinks of Romeo, who, as a Montague, is hated by her family. She implores him to give up his name:
Juliet: O Romeo, Romeo! wherefore art thou Romeo?
Deny thy father and refuse thy name;
Or, if thou wilt not, be but sworn my love,
And I'll no longer be a Capulet.
She believes that their love transcends their family names. She wishes that the feud between their families didn't exist, so that they could be together. Juliet believes that they could be perfect together if it weren't for their families.
At this point, Romeo finally reveals himself, saying:
Romeo: I take thee at thy word:
Call me but love, and I'll be new baptized;
Henceforth I never will be Romeo.
Romeo shows that he is willing to give up anything for Juliet. This sacrifice of his name mirrors his later sacrifice, his life. His name seems like a simple thing to give up at this time, but both Romeo and Juliet later realize that their love comes at a higher price.