Why can't single LEDs produce white light directly? Why does production of white light using a LED require combining a short wavelength LED such as blue or UV, and a yellow phosphor coating?
Why can't a single LED produce pure white light?
 A: At its heart, an LED is a diode fabricated out of semiconducting materials.  There are fixed bandgaps for a given material (or mixture of materials at a given ratio), so only certain energy states are available.  This means only certain wavelengths of light can be generated, since wavelength is a function of photon energy.
In theory, if you could design some  diode which had three different bandgaps, you could produce, perhaps, Red, Green, and Blue output to achieve apparent white light.  In practice, I doubt this is achievable.  You could build close-packed clusters of RGB diodes -- think LED television sets-- but I believe the color-temperature and efficiency of a LED + phosphor design is much better for general lighting purposes.
A: Actually, there is a type of LED that does directly produce white light. It was developed in the early days of blue LED development (mid 1980s) and consisted of a diode in zener breakdown and a blue lens. A diode in breakdown produces, for lack of a better word, light noise (white noise = white light). I doubt that these type of white LEDs are available today simply because they are too expensive and they consume too much power. When on, they have a forward voltage of around 7 volts (breakdown voltage) and have a "reverse" breakdown of about 0.6 volts (forward biased diode). 
A: White light does not have a single frequency, it is a mixture of all of the colors. Similarly pink does not have a single frequency it is a mixture of red with white i.e. white with extra red.
If the color does not appear in the color spectrum (rainbow) then it is not a single frequency but a mixture of different frequencies.
In an LED light is produced by an electron dropping from an excited state to a relaxed energy state which produces a light of a given energy (frequency) however all of the light is going to be of the same color since the bandgap (energy difference) is going to be the same for all of the electrons.
It is possible (see here) to put three different LEDs on one chip corresponding to Red Green and Blue. This produces the effect of a white light. But is not truly white since it only contains Red Green and Blue and no Orange Yellow or Violet / purple.
Your computer screen cannot produce true whites either for the same reason. If you see yellow on your computer screen, there is not light of the frequency yellow only the right amount of Red Green and Blue to convince your eyes nerves to fire the same as if it were a true yellow frequency.
A: If we define "pure white light" as the combination of all the frequencies if the visible spectrum, then the possibility of one diode generating it, is practically nil.  If we define "white light" as the combination of 3 frequencies (red, green, & blue), then we can generate it with 3 small and "closely spaced" LEDs with the appropriate color and amount. I am going to call this method the "parallel" method of generation. In theory, it should be possible to create a diode with 3 different(but appropriate) breakdown regions, on "top" of each other (serial method), to achieve the same thing.  But regardless of the method used, the minimum number of different frequencies required to produce "white light," is three.
So, to sum up, white light requires a minimum of 3 different frequencies, a given diode breakdown region can only produce one frequency, therefore white light can not be produced by a single breakdown region (single LED)! 
A: Fun fact: If you over-power a LED, it will start to heat up and possibly glow, thus producing a Blackbody spectrum = a Planck spectrum. Crank the power high enough and it will glow brightly, i.e. produce a white spectrum directly. Don't touch it though, and mind you the LED will be destroyed in the process ;) (Hey, this is a physics forum after all ;) )
