If the reader has earlier labelled Mersault as an anti-hero and as possessing an apethetic nature, the final chapter offers little to change this assessment.
While Mersault contemplates a sense of hope of "escaping the machinery of justice," he follows this thought with a return to cold reason.
"Of course hope meant being cut down on some street corner, as you ran like mad, by a random bullet. But when I really thought it through, nothing was going to allow me such luxury. Everything was against it; I would just be caught up in the machinery again."
In this chapter Mersault demonstrates his passions, shouting at the chaplain. This episode can be taken as evidence against apathy (but there is evidence throughout the novel that Mersault is not apathetic, per se, as much as he is highly selective in the things he cares about).
The final chapter takes up a theme of certainty which is expressed in Mersault's consideration of the "arrogant certainty" of the mechanisms of justice after the verdict is delivered. Additionally, the chaplain expresses a certainty in his own perspective regarding religion and life after death. What brings Mersault into a fit of rage is an expression that runs contrary to the chaplain's and which is painfully constricted by the absolutism of the justice system.
He speaks to the truth of his own philosophy, which is closely associated with Camus' theory of the absurd.
"I had been right, I was still right, I was always right. I had live my life one way and I could just as well have lived it another. I had done this and I hadn't done that. I hadn't done this thing but I had done another. And so? It was as if I had waited all this time for this moment and for the first light of this dawn to be vindicated. Nothing, nothing mattered, and I knew why."
People die, yet they seek to live life as if there is no end. Presumptions of eternal law or of an afterlife are attached to a sense that the limits of mortality has been overcome by morality and this, for Camus, is evidence of an absurd point of view, positing a belief in two mutually exclusive things at the same time. It is absurd to believe that life is both finite and infinite (and the infinite is implied in concepts of eternal law or apriori moral codes). There is a suggestion in the final passages of The Stranger that the moral sentiment Mersault feels foisted upon him by others is also a device or means to help individuals create a basis for undoing their limitations -- for living, absurdly, as if death were not a final limit.
"Camus and other thinkers describe this situation as “the absurd.” Human beings—seekers of meaning in a meaningless universe—live in a condition of absurdity" (eNotes).
Thus the moral feeling that functions as the fundament of moral values that Mersault chooses to stand against is depicted as a rather arbitrary and willful system that ignores a basic principle of life -- that it ends and thus the value of life is life itself, not a morally determined relative weighting of right, wrong, sympathetic or cruel.
For Mersault to choose a perspective that side-steps conventional morality in this way, he must be able to see himself as existing outside of the bounds of conventional moral thinking. This is part of the formal irony of the text as Mersault thinks outside of conventional morality but is condemned to death by the same system of sentiment that he eschews. He lives and dies in a society that fully embraces a sense, like the chaplains, that moral sentiment is eternal and/or derived from a source beyond the individual.
Is the philosophy that Mersault represents apathetic or is it instead dissenting or rebellious? His speech with the chaplain shows that Mersault is invested. He is not detached from all things. He is, rather, detached from a conventional morality that he has resisted throughout the novel.
In the context of the narrative (featuring a man who loses his mother, falls in love, kills a man and is condemned to death and who feels no remorse), the protagonist has no enemies and has no clear conflict to resolve. The conflict is not between Mersault and any other characters, not even the justice system. The conflict instead is between one point of view and another, with Mersault functioning as a case example of a man persecuted, in large part, because he fails to adhere to socially accepted norms of moral feeling.
"When society condemns him, Meursault realizes that he is not being condemned for taking a human life but for refusing to accept the illusions society promotes to protect itself from having to acknowledge the absurdity of the human condition. In effect, Meursault is condemned to death for failing to weep at his mother’s funeral" (eNotes).
The heroic or anti-heroic qualities of Camus' protagonist therefore must relate to the conflict at hand, which is not a physical or even emotional conflict but a philosophical one. Is Mersault a philosophical anti-hero? What would this mean?
Mersault is not presented as a nihilist. He does not argue that human life has no value. Instead he argues that the feelings people invest in relationships are often used to create false values, evaluating and grading a spectrum of behavior in accordance with a socially-selected set of codes.
One can certainly read this challenge to conventional moral thinking as an antagonism and read his actions (failing to mourn his mother then killing a man and remaining unrepentant) as contrary to a sensible system of values held by his society (and others). While Mersault's character does not suggest a position of anarchy, he does present the possibility of radically rethinking a moral system whose essential purpose is to protect a sense of meaningfulness in life while also protecting human life and property.
To rebel against the foundations of morality is to upset the structure of law that keeps men from killing one another and stealing from one another. In this light, Mersault may be an anti-hero, yet if this is the case it is hard to see him as also being apathetic.
There is ample reason to see Mersault as neither apathetic nor anti-heroic and instead see him as a figure attempting a radical honesty that leaves him open to attack from a society that is suspicious of unconventional thinking.