Perhaps the easiest way to do this, would be to connect your Galaxy Nexus to your computer and use a 3rd-party tool to check the file system for errors.
Please bear in mind that using this method you will only be able to check the mountable storage, and only if your device is in USB mass storage mode.
At current, I know of no way to check the internal partitions for errors - this does not mean there isn't one, however.
Even if you only have an internal SD card, then you should still be able to mount it as USB storage - something has to act as an SD card, for the camera app to function. Something is mounted to /sdcard, and this is what is mounted when you enable USB mass storage.
Ok, last but not least - do you have a recovery installed? If so, you can use this flashable zip to do what you want. If not, then are you rooted? You MUST be rooted to be able to this, as it would involve copying a binary to your system.
Ok. This CANNOT be done while the device is turned on, in user mode, as the partitions must be unmounted to scan them. This is why that zip is that - a zip. If you cannot flash the zip, there is no other way unless you can find a way to unmount the filesystem's when they are being used.