At the start of the chapter of Science and the Modern World where he turns back toward philosophy proper, he comes close to a real characterization, of what he means by rationalism, though not of rational:
Philosophers are rationalists. The are seeking to go behind stubborn and irreducible facts: they wish to explain in the light of universal principles the mutual reference between the various details entering into the flux of things. Also, they seek such principles as will eliminate mere arbitrariness; so the remainder of things shall satisfy some demand of rationality. They demand meaning.
So in some sense, complete coverage without compromise is the goal, and that includes compromise with the arbitrariness of terms and the fitting of concepts into narrow molds that has historically been required by science of itself. (This is especially true now that science itself has found those molds do not accommodate it.)
(To your point about Greeks: Whitehead takes the position very early in that same chapter that we cannot define philosophy in terms of antiquity. We simply no longer have access the same thought processes as ancient people, we lack the context to actually recover them, and can only address their writings from our own perspective, which is inescapably different.
He invests some time in addressing the notion that science is overly rationalistic, blaming the misinterpretation on the Romantic Movement.)
Science does not buy into this agenda wholeheartedly, for the same reason Protestantism and the counterreformation keep Western religion from really doing so anymore or ever again: because it is, at root, based in a faith that cannot be rationally justified, and it does not care. To that degree, it is antipathetic towards rationalism in this sense.
Pressed for philosophical justification for having faith in induction or statistical inference, scientists as a whole simply fall back on success as proof. Success is not justification, it is simply success.
When that is criticised deeply enough they back off to claiming that science is not an attempt to explain, but only to predict, because then success would be all that matters. Yet we use it to explain. We do not simply amass data that we then draw correlations out of, we make models and presume that relatively few verifications of the model make it reliable, just because the model itself has an intrinsic mathematical structure.
If the goal were simply to predict, we would leave it at raw statistics and not contrive theoretical mechanisms. And when we did use theories, we would not think of them as intrinsically meaningful, which we most certainly do. They would have to be simply shorthand summaries of the observed correlations.
At its base, faith in statistical inference simply has to be taken as a fact, and it cannot be rationalised. We need to believe that our experience of the regularity of past interactions with nature means it is basically regular, and that it somehow actively resists 'miracles'. Otherwise, we could equally explain success by being on the right side of God, just very lucky, or deluded. But no real proof of that assertion is possible.