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. Also, a free electron can emit a photon, provided it absorbs one first (and the a photon is of the same amount of energy that it originally absorbed). 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 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).