When Romeo states, "This day's black fate on more days doth depend," he could mean that future events will be greatly affected by these events that have taken (and will take) place on this day of Mercutio's death. However, I think it is more likely that he means that just how "black" today's events are will be determined by the future events that result. The import of the day's events will grow as more events unfold as a consequence. He uses imagery and metonymy when he describes fate as "black"; the color black, in Western cultures, is often associated with sadness, evil, mourning, and loss. When Romeo says, "This but begins the woe others must end," he seems to suggest that the pain and sadness begun on this day of Mercutio's death will continue for many days to come.
Of course, such a statement foreshadows all of the terrible things that will happen and continues the motif of fate that runs throughout the play. (A motif is a recurring symbol that typically leads to a theme. In this case, the theme could simply be that some things are just not meant to be.)