Why is it that only ozone absorbs UV radiation?
The ISO 21348 Definitions of Solar Irradiance Spectral Categories defines UV radiation as that portion of the electromagnetic spectrum having wavelengths from 10 to 400 nm.  Different atmospheric compounds absorb more or less strongly than others throughout the UV portion of the spectrum.  

The following table was reproduced from this Wikipedia article:  

  

Although not fully quantitative, this table indicates that nitrogen, oxygen and ozone all absorb different portions of the UV spectrum.  Water vapor is also an important UV absorber that is not discussed in the above table.  

Also, regarding the UV absorption of (liquid) water and your comment that "It is widely accepted that life on Earth wouldn't be possible without ozone layer", it was in large part the protection from UV radiation afforded by liquid water that allowed aquatic organisms to survive prior to the existence of the ozone layer. If the ozone layer were to vanish today, although the nature of Earth's biosphere would change drastically, I suspect many aquatic species, particularly microbes, would adapt and continue to survive in the absence of an ozone layer. It would be bad news for you and me though ;)