Only the text and the radio button of closing reasons are clickable. However, a full-width clickable area would be easier to hit and less frustrating. IIRC, this was also possible in the past.
This seems to have been implemented and it's currently live. Not sure when it was enacted, though (it's not in Recent feature changes to Stack Exchange as far as I can tell), so maybe a team member can include it in its proper spot there?