The question of blame does, indeed, lie with one's interpretation of Fate as mentioned in the Prologue's "star-crossed lovers." That is, at what point does Fate enter Romeo's life? If it dictates his encountering Juliet at the party and is not a deliberate choice by him, then Romeo's accountability for his actions is mitigated by this first fateful meeting. Or, if "fate" enters later into the actions of Romeo, he must be accountable since he has taken the initial action of approaching Juliet with his love.
As Robert Frost writes, the choosing of the one road, makes all the difference, for all subsequent actions and occurrences are a result of that first choice.
At the beginning of Act III, Mercutio does not know that Romeo and Juliet have married and so he still holds the grudge against Tybalt. Romeo does not want to fight, but once his dear friend, Mercutio is killed by Tybalt, he feels he must defend his friend's life. He realizes, as he says in lines 111-117, that Mercutio has died on his behalf and that his love for Juliet has made him effeminate, meaning his love has made him not as manly or courageous. Once he realizes this, he becomes angry and decides to avenge his friend's death, a death that Romeo believes is actually his fault since Mercutio fought Tybalt when Tybalt originally wanted to fight Romeo from the beginning (Remember that Tybalt wanted to fight/kick out Romeo from the Capulet party back in Act I, Scene v, but Lord Capulet made Tybalt leave Romeo alone.).
As far as whether or not Romeo is to blame for his unhappiness, perhaps that's up to discussion. Some would say yes since he chooses to keep his love of and marriage to Romeo a secret. What would happen if Romeo & Juliet simply told everyone the truth? Some would say Romeo is not to blame for his unhappiness if we believe what Shakespeare presents about fate. From the Chorus at the beginning of the play, Romeo and Juliet are said to be "fated" to die, and even Romeo, in Act III, says, "This day's black fate on moe days doth depend; / this but begins the woe others must end" (3.1.121-123).