You can't do it in the watch. There are programs that allow you to modify the resulting .fit/.tcx/.gpx file, and you can upload that to Garmin Connect so that the correct information is preserved there.
I just looked at the .tcx file and it's so not idiot proof, is there an easier way to figure it out? I deleted the segment from my Garmin 405, but the info still gets uploaded into Connect. Thanks!