Physics actually was one of my favorite classes in high school as I shall now hopefully show.

The main point as to why I think that this is possible is that the Moon was created by the Sage of Six Paths, that means that there was a time in their world's history where it didn't have a moon and it wasn't destroyed by tidal waves and other tidal forces involved when suddenly losing or gaining a moon. As for dealing with all the fragments that would be impacting the earth, I don't really have a solution to that but this is the fate of the world so who would really care about a few continent sized craters and global dust clouds, a few wind and earth style jutsu would fix both of those .

Now on to the physics, (I will explain my work in the spoiler and these numbers are relative to our Earth and Moon), The amount of Work required to blow up the Moon is around 4.14E28 joules. However the Work required to create the Moon is around 4.52E30 joules which is around 100 times more Work then needed to destroy it. Since we know that the Sage of Six Paths created the moon that means that there was at least one person capable of jutsu that have both these huge amounts of energy and ability to work in space. So saying that there is one ninja in the centuries that have happened since the Sage of Six Paths who is one-hundredth the power of the Sage doesn't seem to unreasonable.

Spoiler Alert! Click to show or hide

This is all based off the equation Work = Integral(Force dRadius) and F=ma

The Work to blow up the Moon would be its Mass X Gravitational Acceleration, however since the Moon is being blown up its Mass and Radius are changing so we have to account for that in the Integral.

So The Moons changing mass would be its original Mass of 7.3477E22 kg times the percent of it that is remaining, I used its volume because everything needs to be in terms of R. (I know that this assumes that the Moon as a uniform density and that the really Moon doesn't but this is an approximation and it averages out for the whole Moon)

So the formula for the Mass of the Moon comes out to be 7.3477E22((4/3 π (1737E3-r)^3)/( 4/3 π (1737E3)^3).

Gravitational Acceleration = GM/R^2. G is 6.67E-11 and M will be the same as above. R is the current radius of the Moon so it will equal 1737E3-r

The final formula for the Work to blow up the moon comes out be the Integral of M((G*M)/(1737E3-r)^2) for r between 0 and 1737E3

Plugging in those numbers gave me 4.14628E28 Joules

The Work to pull the Moon from Earth's Surface to its current distance is much easier to calculate. The Mass will be constant this time at 7.3477E22. For the Gravitational Acceleration G doesn't change, M is the Earth's Mass of 5.9736E24 and R will be the distance from the Earth's core

The formula to create a Moon comes out to be the Integral of 7.3477E22((G*5.9736E24)/r^2) for r between 6.37E6 to 3.9E8

Plugging in those numbers gave me 4.5203E30 Joules