Rotate snapping a bone between two vertices

The way to do this is via bone constraints. The two constraints you are looking for are copy location and damped track. Both of these constraints can accept a mesh target, and you can specify vertex groups for the targets, then mark individual vertices by belonging in those groups, to copy location or "rotate to" particular vertices:

I've assigned the center vert of Suzanne's left eye to a vertex group named "From" and the center vert of her right eye to a group named "To". Then I copy the location of From and damped track To.

There are alternatives to either of these constraints. Notable would be a "shrink to" constraint, which it sounds like you might be looking for. This doesn't just rotate to point the bone at the direction of the target, but actually stretches the bone as well.

These are dynamic, which is to say that as the mesh moves, as the vertices (or vertex groups) change, the action of the bone changes. If you want the bone posed like this, but not dynamically, you can "apply visual transform" from pose mode's ctrl A "Apply" menu, then delete the constraints.

If you want this snapping to be in the bone's rest pose rather than posed pose, you can "apply selected as rest pose" from the same menu.

IF you are animating this process you can create a IK bone constraint on the bone. Snap the 3d cursor to the v0 location Add a second bone in edit mode to be your target for the IK constraint. "target" Snap the "target" bone to the vertex1 (edit mode on the cube, snap 3d cursor, pose mode snap target bone to 3d cursor).

Now you have a min-snap rig- that will drive your bone you want to animate and you can bake the animation down when you are done or leave the IK constraint on.