Editing public profile listing view template
You can simply set a field's Visibility setting to Expose Publicly and it will not use the field as a search function.

From the Visibility field helper text:


  When visibility is 'Expose Publicly and for Listings', users can also
  click the field value when viewing a contact in order to locate other
  contacts with the same value(s) (i.e. other contacts who live in
  Poland).