In Letters from Iwo Jima, there's a quote from James Bradley (the writer of Flags of Our Fathers), who says...
Americans have always taken casualties very seriously. When the number of casualties is too high, public opinion will boil up and condemn an operation as a failure, even if we get the upper hand militarily. Kuribayashi had lived in America. He knew our national character. That's why he deliberately chose to fight in a way that would relentlessly drive up the number of casualties. I think he hoped American public opinion would shift toward wanting to bring the war with Japan to a rapid end.
...Kuribayashi was successful in inflicting a high number of casualties, and the war ultimately was brought to a rapid end. Of course, Kuribayashi was presumably hoping for the war to be ended with some kind of equitable peace agreement, not by America launching two nuclear bombs on Japanese cities.
Did Kuribayashi's strategy work too well in shifting the American frame of mind toward winning the war by any means necessary? If Iwo had been captured in less than a week as planned, how might the rest of the war been different?
There is no clear connection between Iwo Jima and the use of the atomic bombs that I can see, because there is no clear connection between fears of casualties from an invasion and the use of the atomic bombs. The idea that the atomic bombs were only dropped in order to avoid a bloody invasion is a complete after-the-fact myth, one meant to make the dropping of the bombs seem like an act undertaken in last resort. It's not true.
Some additional points:
It wasn't at all taken for granted by the military, or the President, that Iwo Jima would be a model for the invasion of Kyushu. Many members of the Joint Chiefs thought things like Iwo Jima, Okinawa, and Leyte were anomalous cases, and that actual casualties would be lower for Kyushu.
The exact same pressures regarding the use of the atomic bombs would have been there without fears of large casualties. The decision path that led to their use did not really intersect with the casualty question.
The actual plan was always to use the bombs and invade. Not an either/or decision. In fact that military and the politicians were quite surprised when the war ended as fast it did. Even the people who planned out the use of the bombs (like General Groves) did not expect two atomic bombs to end the war.
American public opinion had nearly nothing to do with the use of the atomic bombs. The only bit of the "nearly" that matters is perceived sense of public opinion, but even that didn't really factor into the decision-making.
In general, people assume far more structured or rational decision-making about the question of whether the atomic bombs would be used than actually occurred. This was essentially taken for granted. The major decisions were about how to use the bombs — what kinds of targets, etc. I don't see these as being informed by the casualty question in the slightest. So I don't see that they'd be any different.
There is more that one could say about this, but I hope the above gets at the essence of it. For further reading, Gordin's Five Days in August is a nice discussion of how surprised the US was that the bombs ended the war when they did, and Hasegawa's Racing the Enemy is your best all-around events of the end of the war (including atomic bombing decisions) book.