I don't think he killed her in a literal sense.
Metaphorically, he killed the relationship... emotionally killing her.
The lines "i thought i had a pebble in my sock
i pulled it off and shook out a wasp
it stumbled out lost, and without a pause
unstung as i was, still i stomped it"
refer to him "killing" the relationship even though he didn't get hurt. The relationship is the wasp because it has the potential to be painful so he "stomps it" to avoid it.
I don't think he killed her in a literal sense. Metaphorically, he killed the relationship... emotionally killing her.
The lines "i thought i had a pebble in my sock i pulled it off and shook out a wasp it stumbled out lost, and without a pause unstung as i was, still i stomped it" refer to him "killing" the relationship even though he didn't get hurt. The relationship is the wasp because it has the potential to be painful so he "stomps it" to avoid it.
At least that's what I got.