I think that we can ascribe this to poor (or at least unclear) writing. The original screenplay seems to act like the black hole is within Earth's solar system, yet huge enough to have an effect on nearby stars(!). No mention is made regarding the effect on Earth, nor of the black hole's fate.
EXT. SPACE - CONTINUOUS
The JELLYFISH TILTS, earth receding far behind it -- Spock veers to
avoid being blown apart -- Nero's ship gaining -- firing --
EXT. NARADA AND JELLYFISH - CONTINUOUS 226
SMASH! THE JELLYFISH EXPLODES, SPOCK SEEMINGLY KILLED -- AND IN THE
BLAST, THE RED MATTER IGNITES! A MASSIVE BLACK HOLE STARTS TO FORM,
THE BIGGEST ONE YET! STARS DRIFT TOWARD ITS CENTER, PULLED INTO THE
SPHERE BY THE GRAVITY WELL AS IT STARTS TO EXPAND --
In the film's official novelisation, written by veteran sci-fi author Alan Dean Foster, almost the exact opposite occurs. Spock is described as having taken the ship out of the solar system (and into deep space) and the black hole is described as being a small, localised phenomenon which is heading away from Earth.
“The plasma drill has been severed and the platform has crashed into the surface!” This news was followed by an even more startling report.
“Ambassador Spock’s ship has been stolen and is heading outsystem!”
On screen and behind them, the last vestiges of the warship Narada
collapsed inward and upon themselves as they passed the gravitational
point of no return and vanished into the mini–black hole.
Nothing. The ship’s position relative to the system-departing anomaly
did not change.