Your eyes are not sensitive to UV light so any absorption there will be invisible to you. The visible spectrum you give has fairly uniform absorbance across the visible. The whiteness you see is primarily due to scattering of white room lights or sunlight off the powdered zinc oxide. Light is scattered when there is a change in refractive index between two mediums, in this case air and zinc oxide. The effect is enhanced when the crystals are of a very irregular shape and so have many different scattering surfaces. Sugar and salt show the same effect when powdered even though large crystals of each of them can be made that are clear and colourless. As the oxide has a more or less constant absorption no wavelength is very much more absorbed than any other, or alternatively, all wavelengths are equally scattered.
If you can find a solvent with similar refractive index to the oxide (and in which it does not dissolve) then the crystals will become far more transparent and effectively disappear. It is easier see this effect if you wet a piece of frosted glass with water as it will then become far more transparent. Alcohol or benzene would be better as they have refractive index closer to that of many types of glass. Zinc oxide has a value of 2 higher than any common solvent, such as diiodomethane approx 1.7.
Your impression of color is a result of physical (absorption spectrum of the substance, power spectral density of the illuminating light), biological (cone responses) and psychological (surround, adaptation, fatigue...) factors. From a purely physical point of view the posted spectra are not of a white substance (which would absorb light equally at all visible wavelengths) but of a bluish one as it absorbs less at the shorter visible wavelengths than at the longer ones. And yet you see it as white. This is because your brain averages everything in the scene and sets its white point based on that average much as the automatic white point setting in a video camera does. Thus a piece of white paper looks white when viewed under tungsten light (Coordinated Color Temperature 2400 K) as it does in shadow on a sunny day (CCT 8000 K).
There are three color receptors in the eye and their responses are approximately bell shaped curves located in the short, medium and long wavelength parts of the visible spectrum (400 - 700 nm). If those three are approximately equally stimulated by an object (or, more precisely, if an object stimulates them approximately in the same proportion as the scene) that object will appear white. Note that the spectrum of a white object does not at all need to be uniform or even approximately uniform in order to produce an impression of white. When we look at a TV, for example, what we see as a white screen is actually stimulating our eyes with three rather narrow band clumps of light energy corresponding to the primaries of the screen. It can all seem rather complex. If absorption of visible light occurs in a part of the spectrum (near 500 nm) where all three color matching functions are small and cross over one another one would not notice much change in the color of the material as each of the tristimulus values would change little. But if the absorption were at, respectively, 425 nm (peak of the short wavelength cmf) or 600 nm (peak of the long wavelength cmf) the, respectively, short or long tristimulus would change appreciably and the effect be noticeable.
The key word in your question is "appear" and the answer to it is "yes". We can go straight from your posted spectra to color coordinates in any of several systems and I'm confident that by playing around with illuminants and choosing absorbtion peaks properly we could find spectra distorted relative to yours that were imperceptibly close to the same color as the one yielded by the original spectra. These are called 'metamers'. Metamerism (i.e. producing something of one color that looks like a desired color even though the spectra are very different) is obviously the basis for color TV, color printing etc. In fact we could photograph your zinc oxide and look at it on a TV screen. It would still look white though the light coming from the screen would be missing several chunks of wavelengths.