and off he goes to what he already believes will be his death.

What Romeo says, however, is not necessarily what Shakespeare wants his audience to believe.  We, as auditors of the play, need to evaluate action as well as words: what do these characters do, despite what they say?

And if you look at Romeo's behaviors (not to mention the behaviors of virtually every other character in the play including Friar Lawrence), you can see that he consistently makes choices in haste, without thinking-- choices that lead him to his death not because "Fate" willed it so but because Romeo's actions made it so.

I cannot think of a single place but one where Romeo ever attempts to forego impulsive and impetuous action, and that one place (3.1 when he at first attempts to stop the conflict between Tybalt and Mercutio) catapults him into a totally impulsive act: slaying Tybalt.

Since you asked if the statement above is valid, I'll have to say that no, the question isn't really valid, or at least, it isn't that useful. Why? Because both statements are true. There are strong elements of fate in their deaths (the "star crossed" factor), most notably the fact that these two came from different and warring families. (There were some other factors, but that is the main one.) However, if they hadn't made choice after choice as they did, they would have lived, and might have even been allowed to marry (unlikely, but possible.)