The Remesh modifier is made for modeling the surface only. Despite the look could appears quite similar, they are not voxels. It's probably easier to look for another workflow such as...
Particle system - Grid
From the manual:
Particles are set in a 3d grid and particles near/in the elements are kept.
The emission algorithm automatically kill the particles outside the mesh (it must be watertight for correct results). Then you can just duplicate a cube in each particle's location as shown in the picture below.
In order to get the cube size that allows no gap between faces you must take into account the reference object bounding box's size and the grid resolution parameter.
In the provided example, for a cone that stays inside a cube of 1 unit and a resolution of 100 particles, the cube size is 0.01 units.
And here's a Suzanne made of cubes splashing on the floor:
EDIT: ADDED BY OP
For those who can't figure out how to make those particles behave well physically:
Is there a way to make particles behave like rigid body objects?
Look at the answers by WChargin and MarcClintDion.
In a nutshell, the generated blocks are particles internally, so they will always behave like points, dimensionless. But we want the boxes to behave like real cubes, don't we? So, we extract (generate) real objects from those
duplis, by doing this:
Now that they are independent objects, we enable physics on one object and copy it to all the rest by doing this:
By the way,
If, for example, you are making them fall on a floor and letting them scatter away, some might pass through the plane, as in the OP's case. So you should increase the (floor) plane's collision margin until no cube passes through.
If it is apparent that cubes are not actually touching the floor due to increased collision margin, then you can change the plane's origin to make them look like they are really hitting the floor.