Google - What happens if I don't pick up this household bug I just killed? Mutation, Decay? Do I?

You don't need to; nothing incredibly bad will happen if you don't. It'll either decay or be eaten by another bug. If it's easily visible in your home, I recommend picking it up. If so and you still don't want to touch it, you could try using a dustpan to sweep it up, or if you don't have one, you could just sweep into an area where it is unseen.