The crystalline state of water ice in the Solar System depends on the temperature history of the ice and the influence of energetic particles to which it has been exposed. We measured the infrared absorption spectra of amorphous and crystalline water ice in the 10-50 K and 10-140 K temperature ranges, respectively, and conducted a systematic experimental study to investigate the amorphization of crystalline water ice via ionizing radiation irradiation at doses of up to 160+/-30 eV per molecule. We found that crystalline water ice can be converted only partially to amorphous ice by electron irradiation. The experiments showed that a fraction of the 1.65 mu m band, which is characteristic for crystalline water ice, survived the irradiation, to a degree that strongly depends on the temperature. Quantitative kinetic fits of the temporal evolution of the 1.65 mu m band clearly demonstrate that there is a balance between thermal recrystallization and irradiation-induced amorphization, with thermal recrystallizaton dominant at higher temperatures. Our experiments show the amorphization at 40 K was incomplete, in contradiction to Mastrapa and Brown's conclusion (Icarus 2006, 183, 207.). At 50 K, the recrystallization due to thermal effects is strong, and most of the crystalline ice survived. Temperatures of most icy objects in the,Solar System, including Jovian satellites, Saturnian satellites (including Titan), and Kuiper Belt Objects, are equal to or above 50 K, this explains why water ice detected on those objects is mostly crystalline.