An alternative viewpoint:
The “law of inertia” (which can be stated alternately as the ‘conservation of momentum’) can actually be immediately inferred from the hypothesis that space is homogeneous.
In other words, the “law of inertia” is a direct consequence of the fact that space itself ‘looks’ the same at every place in the space-continuum. Thus, in the absence of other objects, a given object has no way to tell where it is in space, because everything looks the same–furthermore, it doesn’t have any way to tell if it is moving (if you were floating by a perfectly clear, flat piece of glass, but couldn’t touch it, could you tell how fast you were going? or if you are even moving at all?). If the object has no way of knowing how it is travelling through ‘space’, how could it alter its path.
This, of course, isn’t a “proof”, but it’s one way that theorists might think about it. Well, at least theorists who ignore relativity and quantum physics.