Well, it is possible, that it starved to death, but more than likely a different issue. I have a Starry Blenny (which is very closely related to the Lawnmower, unless I'm mistaken) who was only eating what he could find on rocks and glass... Over time he started chasing any food I put in the tank, and now he eats like a pig.
Every fish is different. Some will eat Nori, mine won't. Some will eat flakes/pellets/mysis/etc. Mine eats all of the above, but won't touch nori, and now he doesn't even graze the glass or rocks, he just sits his fat butt on the Monti shelf, waiting to be fed.