To answer your question, let's first look at the semantics of your example sentence: Somebody has done something unforgiveable, and now his words can neither undo ("ändern"), nor justify ("rechtfertigen") his actions. Notice that these things are not equal in value: Undoing ("ändern") an action is more than merely justifying it ("rechtfertigen").
The phrase "oder auch nur" serves to express this difference in value. To understand why we need all of its parts, let's take a closer look at them individually:
The conjunction "oder" ("or"), in its solitary form, expresses that there are two alternatives of equal value:
Nächstes Silvester laden wir unsere Freunde ein oder feiern allein. (Was von beidem wir tun, ist mir egal.)
Coming New Year's Eve, we're going to invite all our friends or celebrate on our own. (I don't care which of the two we'll actually do.)"
For our purposes, however, a solitary "oder" won't do because our two alternatives aren't equal in value:
Es gibt nichts, was er sagen könnte, was die Tatsache ändern oder rechtfertigen würde.
There's nothing he could say to change or justify the fact.
To add this additional meaning, we need to add the adverb "nur" ("at least" in this context) to the sentence:
Es gibt nichts, was er sagen könnte, was die Tatsache ändern oder nur rechtfertigen würde.
There's nothing he could say to change or at least justify the fact.
Note that this construction already expresses everything the speaker wishes to impart (i.e. that he deems "justifying" worth less than "undoing").
However, if the speaker wishes to add a sense of frustration and disappointment to the sentence, the particle "auch" (here "even") will serve this purpose:
Es gibt nichts, was er sagen könnte, was die Tatsache ändern oder auch nur rechtfertigen würde.
There's nothing he could say to change or even at least justify.
4. "oder auch?"
Lastly, let's take a look at the last construction you've mentioned in your example - the variant without the adverb "nur":
Es gibt nichts, was er sagen könnte, was die Tatsache ändern oder auch rechtfertigen würde.
To my ears, this sentence does not work for two main reasons. Firstly, as mentioned above, the "or" makes it sound like "ändern" and "rechtfertigen" are equally valuable. Secondly, the "auch" even strengthens this impression. This is because the word "auch" can also be an adverb, meaning "as well".
To sum up, every individual word has a part to play in the given construction. While some, such as "auch", are optional and can be added for emotional nuance, others (such as "nur") are obligatory because they narrow down the overall meaning to exactly what the speaker wishes to express.