If you are the one who made this public profile then you need to delete it from being public first. You can do that by clicking on the globe then on the right side you will see your profiles and around 3 buttons underneath them. One of them is for deleting it I think it looks like an X but if you have tooltips enabled it will show you which one to delete it. After deleting it from being public, you can now delete it from your profiles and click sync settings.