I read that, on other devices, adding the line "qemu.hw.mainkeys=1" to the build.prop file under /system would do it but I added the line and it did nothing. Then -- since I was just beginning to customize a newly rooted Nexus 7 (32GB, in case that matters) I scrubbed EVERYTHING and took it back to stock, then rooted it with WugFresh's Nexus Root Toolkit (Version 1.5.5). Next I entered my sign-in data and installed Root Explorer from Google Play (so I could edit system files). Then -- before I did anything else to the tablet, I again added "qemu.hw.mainkeys=1" to the end of build.prop in /system. It worked! The Navigation Bar disappeared and has not reappeared no matter what I've done (reboot, add apps, customize, etc). Better yet, my wallpaper image instantly resized to fill the screen space I'd recaptured. My rows of icons adjust to fill and use the space and no black bars are left anywhere on screen.
It seems unlikely that I'm the first to discover that this works with the Nexus 7 since I'm not a very sophisticated user but in case this adds value to the XDA forum, which I use and very much appreciate, here it is.