Can a free hydrogen atom constitute ionizing radiation? Radiation is basically just particles flying around, right? Are free hydrogen atoms just typically not moving fast enough to be considered "radiation"?
 A: The key question here is "At what temperature?".
Room temperature corresponds to a mean energy of about 1/40 eV and the tail just doesn't go high enough to be interesting.
Once a significant portion of the atoms have a few eV, then they could begin ionizing some materials but the per atom dose will be tiny and there will be no penetration to speak of. Once they have close to 13 eV a large fraction of the hydrogen will be ionized and you'll have a plasma.
We typically talk about ionizing radiation having at least tens of eV and preferably rather more than that.
A: A hydrogen atom is a neutral object, and even if it somehow acquired large energy it would still not ionize measurably , as a neutron does not ionize. The term ionize means electrons are kicked out of other neutral atoms making them positively charged for a while.
Hydrogen is composed of an electron and a proton. If hydrogen is hit by ionizing radiation and becomes a free electron  and a proton, those  are certainly part of what is termed ionizing radiation, if they can gain from the impact enough energy not to recombine to a neutral hydrogen atom.
Radiation  is a more inclusive term, and encompasses ionizing radiation, but also includes neutral objects with high enough energy to penetrate bulk materials and cause radiation damage from scatterings with nuclei, which then produce high energy ionizing ( charged) radiation products, as neutrons do. Hydrogen is neither a scattering product from such scatters nor easily accelerated to energies high enough to penetrate materials without losing its electron and becoming a proton, nor a byproduct of nuclear decays.
See also the other  answers .
