More precisely, Nietzsche simply "did not enter into a discussion of the differences between ancient and modern philosophical writers on ethics" because he took them to be similar at least in the respects that mattered to Nietzsche. As White puts it: "He believed that Socrates marked the beginning of the decadence that was accelerated by Christianity and brought to a contemptible nadir by the anaemic eglitarianism of modern Europe...[T]he mainstream of Greek ethics reprsented the morality of the herd, bent on suppressing the gifts of splendid individuals" (p. 43).
That seems to me roughly right (though I have not taken the time to look back carefully at his lectures on the Greek philosophers), though it perhaps bears emphasizing that this is not so much a denial of the more prevalent view among German philosophers about the ideal of harmony represented by Greek ethics, but a lack of interest in the topic: even if there were such a difference, it would not affect the Nietzschean worry about ancient and modern ethics which White articulates.
White goes on to note, correctly, that Nietzsche's nostalgia for the Greeks was for the pre-Socratic Greek philosophers and poets (p. 44), but then makes the unusual suggestion that Nietzsche celebrates a different ideal of harmony that he finds, in all of places, in Heraclitus (p. 45). White offers the following quote from Nietzsche's early essay Philosophy in the Tragic Age of the Greeks:
White concedes that "this is not a harmony that enters into human affairs" (45). It is also not, as far as I can see, an ideal of harmony for Nietzsche, as opposed to a tendency he notes in Heraclitus. Put aside the question whether Nietzsche has gotten Heraclitus right; is there any evidence that Nietzsche endorses this notion of harmony? Perhaps when the doctrine of will to power is taken systematically, as John Richardson does in Nietzsche's System (Oxford, 1996), we get a similar notion? Is that the thought?
Everything that happens, happens in accordance with this strife of [opposites], and it is just in the strife that eternal justice is revealed....
Do guilt, injustice, contradiction and suffering exist in this world? They do, proclaims Heraclitus, but only for the limited human mind, which sees things apart but not connected, which sees things apart but not connected, not for the con-tuitive god. For him all contradictions run into harmony.