I realized that I got a downvote, which normally means the answer is wrong, while I think it is true. But the OP has updated the question, so I take this as a chance to also update my answer.
It seems you expect one simply answer why different things in nature appear white, but the truth is there are different reasons for different things.
For example, clouds are white because of Mie scattering, that is scattering of photons on particles (here small water droplets). There is almost no wavelength-dependence of the scattering efficiency in the visible range, so that all wavelengths in the incoming sunlight are scattered with the same probability, and as a result the cloud is white. But notice, the incoming light has to contain all wavelengths for that. During sunset for example, the cloud base is sometimes red because illuminated by the sun, which is red when close to the horizon during sunset (the reason is that blue light is scattered out of the direct beam due to Rayleigh scattering that prefers to scatter blue light and is responsible for the blue sky during the day).
Another white material is milk, which is a liquid. It is also white because of scattering, this time on particles within the milk. But again, you need an illumination of a light source containing all wavelengths to get a white substance. In a green room, a glass if milk will appear greenish of course...
You received some comments for a transparent material above. The reason is that transparent materials don't do any changes to the color, there is no absorption. Photons of all wavelengths are passing through. Well, actually that is not the entire story. Light is an electromagnetic wave and thus in classical electrodynamics we think of small dipoles that are induced in the material (this was the point in my original answer). The dipoles emit secondary waves that usually cancel out each other inside the material, but at the surface this results in some backscatter. This is the cause for reflection in transparent glass windows (up to 4% is reflected). This doesn't matter as long as the surface is smooth, you just look through the window. But when you make a scratch or take a hammer and smash the window in many, many small pieces, the pieces will become white. The reason is that you now have a lot of small glass pieces with a lot of surfaces, and reflections occur on each single of them, which in total amounts to a reasonable backscattering, and because each wavelength is treated the same, the effective color is white. Another example is ice cubes. Fresh ice cubes with smooth surface are transparent, but crush them or damage the surface and they become white. (Again, only if the incident light contains all wavelengths.)
You explicitly ask for paper. I'm no expert on this, might be again some different reason. Paper is wood and you need to do chemical stuff (bleeching) to make it white...
However, the point is, there is not that single reason, but a number if reasons why different things appear white.
In classical physics, light as being an electromagnetic wave is considered to induce small oscillations, i.e. small dipoles, in the material which radiate secondary waves. If the surface is irregular, you get reflection in all directions, meaning if you shine sunlight on it containing all wavelengths, the surface looks white from every direction. This is giving snow its white color, for example (actually, photons even penetrate the snow a bit and are backscattered on each surface within the snow producing what is called diffuse reflection).
I think the related wikipedia article https://en.m.wikipedia.org/wiki/Reflection_(physics) may help your understanding, it has very nice explanations and intuitive plots if you scroll down to "Laws of reflection" and then "mechanism" and "diffuse reflection".