I am not sure I would describe Oppenheimer as really having "Hindu ideals"; he certainly adopted some aspects of Hindu influence into his worldview, but his worldview was very much his own (a mixture of many things, from Ethical Culture to Hinduism to a sort of 1930s Marxism).

His famous quote from the Gita is most likely a reference to ideas about duty and responsibility, and not a criticism of the bombing itself, but a justification of his own apparent abdication of care about the loss of life in war. I do think, at some level, he felt it was if not fated, then required of him to do. That is not, I would add, why he did it at the time — that feels like an after-the-fact justification. His reasons for signing on to the project were complicated but seem to have been more individually personal and psychological, like an overwhelming need to prove his worth to a country that had regarded him as something of an outsider (my sense, taken from Ray Monk's biography, is that a huge theme in Oppenheimer's worldview is his experiences with anti-Semitism as a young man, and his desire to show that he was truly "assimilated" and thus "American").

Oppenheimer never, as you note, renounced the use of the bombings, but in fact defended them until his death. There is a letter that he wrote to his former student David Bohm in December 1966 (two months before his death) that I have always found pretty telling regarding his point of view. It is response to a query about a biography of him being written, and about a play by Heinar Kipphardt about his security hearing which apparently irritated him by making him look regretful about the bombing.

What I have never done is to express regret for doing what I did and could at Los Alamos; in fact, under quite dramatic circumstances [edited to read: "on varied and recurrent occasions"]), I have reaffirmed by sense that, with all the black and white, that was something I did not regret. My principle remaining disgust with Kipphardt's text is the long and totally improvised final speech I am supposed to have made, which indeed affirms such regret. My own feelings of responsibility and guilt have always had to do with the present, and so far in this life that has been more than enough to occupy me.

The last bit, about his regrets, was dropped from the final letter sent to Bohm, and could be interpreted about his very troubled family life. But it could also be interpreted, and this is relevant to the famous footage of him and the Gita, as being about his failure to prevent an arms race, which is what he had hoped to do, and what he thought the bombing of Hiroshima and Nagasaki might facilitate (by making clear to the world the terror that lay in a future nuclear war).

I would not read Oppenheimer's interaction with Truman as a complaint so much as a confession of a feeling of heavy responsibility. He did this quite a lot, and used it not as a criticism of the bombing, but as an indication of the special burden he and other physicists assumed as a result of their work, a burden that required them to work towards a safer world. This is what he spoke of when he said that "the physicists have known sin; and this is a knowledge which they cannot lose" and "if atomic bombs are to be added to the arsenals of the a warring world, or to the arsenals of nations preparing for war, then the time will come when mankind will curse the name of Los Alamos and Hiroshima."

Not everyone thought such "confessions" were worthwhile. Truman apparently took offense, because he had in his mind taken on that "burden" and resented Oppenheimer, a technical man, for assuming it (Truman felt "the buck stops here" in general, but especially with the moral questions of the bomb, which is quite interesting since he played a much smaller practical role than Oppenheimer in it being used, but nonetheless is an interesting ethical model). John von Neumann, another Los Alamos physicist, once wryly and critically remarked that "sometimes someone confesses a sin in order to take credit for it," which may have some truth to it (Oppenheimer's profession of responsibility also was what gave him, specifically, a special voice in these conversations).

Anyway, we can't get inside Oppenheimer's head in any definitive way, but my sense of him, from reading many biographies and going over his papers in the Library of Congress, is that he occupied a very odd position on the bombings. He viewed them as tragic and horrible — awful events. But he also viewed them as necessary events, and not necessarily "special" events of bloodshed (in another angry criticism of the Kipphardt play, he invoked "Guernica, Dachau, Coventry, Belsen, Warsaw, Dresden, Tokyo" as things that, in his mind, made it clear that Hiroshima and Nagasaki were hardly unique in the history of human horrors). The "necessity" for him seems to have been much more than the end of World War II, but about a sort of fundamental direction of human history, and notably towards his goal — which he had even started on during the war, before the weapons were used — to think about how to possibly avoid a large-scale nuclear arms race and the subsequent large-scale nuclear war he thought might follow from it. In this mindset, the more horrible the first use of the bombs were, the better, because it would possibly allow the world to avert a future catastrophe when the weapons were larger (they were already contemplating the hydrogen bomb), more numerous, and in more hands.

Which is a different position than most people today see about the atomic bombs. Today the main categories seem to be "they were horrible and thus shouldn't have been done" or "they were a necessary evil because the ended the war," and Oppenheimer's position was somewhat different: "they were horrible and a necessary evil because they may have prevented a future nuclear war." But in his lifetime he seems to have lost faith in the latter to a great degree, especially after he fell out of power. So if there was any regret, it was about the direction that things had gone after 1945, about the failure of international control, about the apparently unrestrained build-up to the arms race, about the dire dangers that the world seemed to be barreling towards despite people like him having tried to warn them.