To some degree, but unless a very large separation occurred, gravity differences between the pieces would be minor, increasing with distance from the sun of course. But the velocity needed to send pieces our way would be tremendous and require some kind of really massive explosion. The current track takes the comet and any debris from a breakup well above and in front of the earth. An unknown might be some kind of massive CME occurring at an opportune time knocking debris into a different trajectory - another highly unlikely possibility and one that would more likely only push pieces further away from the earth's path.
Any encounter with any exiting debris would be extremely remote; more likely we get a fantastic light show from it.
I recall reading this line (or something very similar) in 'Lucifer’s Hammer'!
Thanks for your answers. There will be nothing we can do anyway but hope and pray.