The significance of the Major's interest in grammar is that grammar is the one part of language we can control. It is part of our monitor language (the other would be our natural language, the one the soldier thought was "easy to learn") and it requires a certain set of rules to be applied. When the soldier said that, naturally to him, Italian was an easy language to learn, the Major understood it as if his lessons were losing control of the Narrator's attention and therefore he had to take another way to teach: Enter grammar. If from now on the lessons switch from casual speaking to attending to grammar, the Major would have more control of the situation.
The consequence on his wife's death to be more ironic is that we can definitely see how, even after her death, the Major refuses to let go. He does not even let go of something as simple as teaching someone how to speak Italian. He still "hasn't got it" that he is not to be in control of absolutely everything that surrounds him. His wife was the biggest message sent to him that he is not the one in charge, and yet, he persists in trying to be.