Is it possible to use the raw accelerometer data from a smartphone as a weighing scale? So, by placing an object on the screen, the app would weigh the object and the weight displayed on the screen.
If it is not possible, could some other function of the smartphone estimate the weight of an object?
No. Acceleration can be used to deduce the mass of an object, but it requires a special setup such as the Atwood Machine. The Atwood machine traditionally relies on timing to work out the relative mass of the two objects, but you could do it with just the accelerometer instead.
On phones with styluses such as the Galaxy Note 2, you could use the so called S-pen's pressure sensitivity to measure weight. But this would require creating some kind of mini tripod or some such device to house the stylus. Theoretically you could measure to within 1024 levels of precision since they say they have 1024 levels of pressure sensitivity on the S-Pen (as opposed to 256 on the Note 1's S-Pen).
It can be derived from the physics formula for mass, m = F/a, (mass = force / acceleration). So you can get acceleration from a smartphone, but you'd have to have a constant F, which would require you to do some engineering to provide a constant amount of force. If you're relying on users to provide this force, you'll run into issues of the force applied not being consistent (without the aid of some standardized hardware).
Don't forget mass and weight are different, but once you have the mass you can easily get the weight.
