Is it professional to have a "cited by" section of a personal website?
For any well established research, they will usually have quite a large number of citations, which would make a "cited by" section would be exhausting to maintain and large beyond meaningful readability.  A better route is to link to an database that actually engages in curating ones citations, such as Google Scholar.

I do, however, sometimes see people maintaining a collection of notable articles in non-scientific media.  These are not normally picked up by citation databases and are valuable for showing that work is having impact outside of the scientific community, as well as (sometimes) providing a nice introduction to the work.