Without the moon at all we would still have ocean tides due to the Sun's gravity , but it would be much less pronounced . Also without precession of the Moon around the Earth axial rotation would be more extreme . These factors mean nothing , because even if the moon shattered as you describe Earth would be bombarded with millions of bits of lunar material that would completely destroy our civilization .
If a huge asteroid hits the moon and disintegrates , it is likely that not only will the tides or the behavior of people who will be affected . Due to the proximity of the moon to Earth , a direct impact of a giant asteroid could mean a direct blow to our planet . If the moon disintegrates on impact, of course , the asteroid (although affected by its collision with the moon ) will continue to us .
I speculated about this before . This would affect people in an emotional way , because people are made up of water , so that they do not affect us is absurd . If you can affect an ocean , I am sure that will affect people , but in a positive way emotionally because people often get cranky around the full moon .
Now , the destruction of the moon would be a deterrent to both our lifestyles and global ecosystems . How I'm not sure , ergo why I find interesting question. Now I 'm going to speculate . I believe that ocean currents will be stagnant , as there no longer tides . The oceans depend on rivers flowing into it , but still the ocean is more, it can mean more bacteria in the water , the fish can get sick . Migration Marine life be affected, traditional fishing sites and seasonal so it can be a blow . Because of this stalemate , I'm not sure what would be proper water swim in a few decades after shooting the moon .
It's fun to think about
You want to know what would happen to the tides? I wouldn't worry about that or anything else any more. you do know that if the moon was destroyed the Earth would die right? the moon doesn't only control the gravitational pull of the oceans, but it protects the earth from the direct sun light that would cook the earth and turn it into a charcoal-ed raisin so-to-speak. We would all be dead in a few months and there is nothing any one could do about it short of evacuating the entire planet. It's to scary to think about , so don't dwell on it it will scary you think about it to much.
Take care and Keep looking to the stars, The truth is out there. ....PEACE....cya....