_Accelerating_ charged particles, electrons or protons, emit electromagnetic waves. This means if they are moving with a constant velocity, or they are stationary, they will not  emit electromagnetic radiation (photons).


An electron in an atom (in an excited state) can emit a photon, and drops to a lower energy level. 
But a free, isolated, non-accelerating electron will not spontaneously emit a photon, because conservation of momentum/energy will not hold.

Also, a proton (in a nucleus) can go through a different radiative process, called [beta decay](http://en.wikipedia.org/wiki/Beta_decay) where a proton decays into neutron with the emission of a positron and a neutrino. And a free proton does not spontaneously decay (as far as all experiments have shown).