Where are you in the play?
I am going to answer your question based on the assumption that you have just read Act I, scene v. This is where they met and fell in love.
Romeo's greatest fear is that Juliet is a Capulet, his family's foe. This is true, so your question seems simple because it is justified. A fear we probably don't see is that she will find out he too is a Montague, her family's foe.
By Act II, scene ii (the famous balcony scene) the fear of her turning away because she knows who he is becomes stifled because he overhears her thoughts discovering she still wants him. Romeo's fear of being caught is justified when the Nurse calls from within.
Romeo also fears in Act III that the families will remain fighting. His fear is justified when Tybalt and Mercutio engage in a battle essentially over Romeo. Romeo can't give away at that point why he and Tybalt should be friends.
Romeo's fears regarding their relationship are justified because of their families' continuing rage and struggle. Throughout the play, up until the end, the families remain at odds. If you watched Romeo's response to the first fights, he didn't like knowing the fight was going on... this happened well before he even knew Juliet.