The simplest and most direct way to answer this question in general is that the novel excelled in its presentation of the complexity and simplicity of human emotion.
It succeeded at illustrating through the affair and the eventual twist in the plot that humans are not "black and white" but that our nature is so comprehensive that abstract ideas such as love, hate, and forgiveness transcend the limits of our control, which is a very realistic theme and one that does not disguise love, nor makes it a fantasy goal of Utopian nature. Love is good, and then it can really suck. That is the basic gist of the story.
We should be sure to clarify between the terms "realistic" and "realist" when we are discussing novels. Asking how we might qualify Greene's novel as realistic will give us a quite different answer than asking how it might qualify as realist.
Insofar as the novel does not seek to fantasize romance (as other novels sometimes do), we can argue that Greene's book attempts to present a depiction of love and adult relationships that matches actual experiences, actual life or "reality." There is no happy ending and, beyond that, there is no clean, cut-and-dry conclusion to the story.
The end of Chapter 1 in Book 5 finds Bendrix narrating the idea that his "realism has been at fault all these years," imperfect or inaccurate because he used to think a novel "has to end somewhere." Yet, the book goes on, relating the experiences Bendrix has in dealing with Henry and navigating his own grief.
By pursuing the aftermath of the romance at the heart of the novel, Greene adheres to a "realistic" sense that a person's death is really only one part of a social process that includes grief, mourning, pain and a period of healing. The social aspects of this process are also depicted in the novel, further reifying the novel's conceit that personal experiences connect people to the world (and do not occur in a social vacuum or serve to disconnect people).
While our primary view of the novel's realistic nature may stem from its staunch avoidance of romanticizing love, we can also include these other formal and psychological ideas in our discussion of the ways in which the novel tries to avoid fantasy and thus reflect life as it is actually lived.
Looking at the question of realism, however, we would find that The End of the Affair would not be a likely candidate for this description with this term.
"...realist writers often addressed themes of socioeconomic conflict by contrasting the living conditions of the poor with those of the upper classes in urban as well as rural societies" (eNotes).
In today's critical climate, the social and political elements of realism are integral to the term's usage. Writers like Dreiser (U.S.A.) and Zola (France) are exemplars of literary realism, addressing living conditions and psychological responses to those conditions as a principal aspect of the novel. Greene's interests are perhaps too metaphysical and too resistant to confident "psychologizing" to be considered realist.