This doesn't work, at least not exclusively (more on that after some background). With stabilization, there are two main ways to do it. You can either stabilize during shooting by keeping the camera steady or after by taking unstable video, counteracting some of the motion and adjusting the image to account for other types of motion.
When you stabilize during shooting, gyroscopes, gimbals and springs are used to cancel out forces trying to move the camera to actually keep it stable. This works because the camera doesn't actually move. Things like optical image stabilization and steadicams perform this function.
Correction after the fact is possible, but harder. When a camera moves, the problem is that it changes what the shot looks like. Both angular movement and perspective can vary based on the actual settings of the camera and not just the movement, so there isn't a direct mapping from movement to correction.
There are two main types of camera movement, the camera can move through space, which results in near objects appearing to move more than far away objects as the relative change is greater. (Think of a block directly in front of the lens vs a mountain in the distance, the block can clear the screen entirely while the mountain doesn't appear to have moved.)
The other form of movement is rotation. This makes the rate of change depend on how zoomed in we are (the focal length of our lens). If we have a very wide shot, then slight vibrations don't make much difference. If we zoom in a long ways to a tight shot however, even the smallest vibrations have a major impact on the shot.
None of this information would be available to the data logging on the smartphone, so a lot still needs to be figured out by analyzing the footage. You additionally would need to somehow keep the smartphone and camera together which a) adds weight, b) adds bulk and c) decreases the ergonomics. The data really isn't that valuable as that information and all the other information we need can easily be gained simply from looking for high contrast areas of the image.
Stabilizing after the fact works by doing edge detection on each frame and looking at how the edges move from frame to frame. These edges provide details about where things are in the scene and allow us to actually recreate the motions of the camera and the objects within the scene by looking at all the changes. If they are consistent overtime with a stationary scene, we know they are static objects, if they don't, we know they are moving objects.
So, basically, the data the smartphone would gather isn't needed, isn't that helpful and adds too much disadvantage in trying to capture it to justify the gathering in the first place. It could possibly save on some post processing load to simplify calculations from edges or provide improvements in cases where the footage lacks good trackable subjects, but generally, I wouldn't expect it to be worth much. If it costs more than $100 or so, you are better off going with hardware stabilization that will provide far more benefit.
For more of an idea about what can be determined about video just from analysing the footage itself, I suggest taking a look at Microsoft Research's Hyperlapse Project. Pay particular attention to the path they are able to derive as that is the same thing a device lack this would provide the rotational and acceleration data would be helpful towards determining.