I don't think a memory is truly alive (In regard to Tom Riddle, I mean.). Something that is not alive cannot be killed by any means. For example, Nearly Headless Nick¹ -- he couldn't die twice because he was already a ghost when he saw the Basilisk. Accordingly, the memory of Tom Riddle wasn't alive to be killed.
‘Justin must’ve seen the Basilisk through Nearly Headless Nick! Nick got the full blast of it
but he couldn’t die again ... ’
Chamber of Secrets - page 215 - Bloomsbury - chapter sixteen, The Chamber of Secrets - Harry Potter
¹I used Nearly Headless Nick just as an example of a spectre-like being and how he couldn't be killed by the Basilisk; a memory isn't exactly a spectre, but it is non-corporeal (Unless it's made corporeal through emotion like Tom Riddle did with Ginny Weasley.). So, yes, I know Tom Riddle was never a ghost in Chamber of Secrets. It's just that a memory isn't corporeal or alive in the true sense of the word.