Your biology teacher's question is unfortunately poorly formulated and the justification is even worse, not just biologically but also logically. For the logical error, consider this statement: Knights wearing chain mail heal faster because they are protected by their armor. This makes no sense. Perhaps if you allow that the initial damage is less, but even that makes little sense.
It doesn't really make much sense to talk about healing speed of white matter and grey matter because you are talking about two different things. White matter is mostly axons and their myelin sheets, whereas grey matter consists of cell bodies and neuropil. Grey matter damage implies cellular death. White matter damage implies loss of connectivity. Grey matter damage can also lead to white matter degradation (i.e., the cells that sent projections are now dead). White matter damage can also lead to grey matter degradation due to loss of connectivity.
However, it is true that different types of brain injury could differentially impact white versus gray matter, but going into all the different causes of brain injury is too broad of a topic for a single question at stack exchange.
Furthermore, repair of CNS tissue is just as much about compensation rather than repair per se. Repair processes are important for limiting damage, cleaning out dead tissue, and restoring some types of connectivity, but recovery from a major injury is just as much about brain plasticity and effectively relearning how to function after an insult.