I am not sure there is an app to accomplish this. As Titanium backup requires root to do just about anything, actually I am pretty confident the app is 100% useless without root. My Backup may work without root, to just backup apps, but I would imagine that the installation method would require you to babysit the device to reinstall them all, clicking ok to install for each app. The only way to root the Galaxy Nexus requires you to unlock, which wipes the entire device, so rooting then using one of these apps isn't really a solution to this problem.
That said you have two options that I can think of.
First: Google syncs your apps to the cloud. By that I mean when you get a new device, a replacement GN or a completely different device, when you log in with your gmail, it should automatically download all of your apps for you. This can take a while depending on your network connection, also this method requires data, which if you are on a bandwidth limited plan this may not be an ideal outcome.
Second: This requires a PC and a bit of setup on that PC.
[GUIDE] Full Phone Backup without Unlock or Root - xda-developers <-- This is what you would be doing, using adb backup to make a full backup and restore. I believe
Nexus Root Toolkit v1.5.3 | WugFresh <-- that can do the adb backup and restore I believe (I have never used the toolkit personally).
Side note, in any, or all of these methods, since there isn't a removable SDcard, you will need to back it up to a PC yourself. So if you have pictures/videos/music/whatever else on there you would like to keep that isn't backed up elsewhere, it may not be a bad idea to back them up.
E2A: I can go into further detail if you want me to.