They would have had another bomb ready to drop by August 17 or 18th. They were, at the time of Japan's surrender (August 14, five days after Nagasaki), preparing to send its plutonium core out to Tinian for preparations, should Truman allow it to happen (on August 10, he had taken control of future bombing from the military, so it was, for once, truly up to him).

There are many documents alluding to the possibility of future bombings, since they were discussed, but no policy was set, so we cannot say with any real confidence how they would have used said weapons. Would they have dropped one on Tokyo, as some Generals recommended? Would they have used it against on of the remaining two targets (Kokura and Niigata) on strike order for July 25, 1945? Would they have saved them up and used them in batches, as Chief of Staff Marshall was contemplating? We don't know, but all of these things were on the table.

It is hard to say what the American public would or would not have supported in the abstract, but postwar polling makes it clear that a majority of Americans thought the use of the bombs was appropriate and a substantial amount wished even more could be dropped. But this response was in part conditioned by the credit that the atomic bombs were given for "ending the war." If the atomic bombs were not perceived to have promptly ended the war, would they be seen as an appropriate activity? No one can say, but their "appropriateness" was largely (but not absolutely) conditioned on their being the lesser of two evils. What if they just killed but did not induce surrender? Well, that would require a different framing.

I have an article on this very topic coming out in National Geographic History this summer... there's lots to say on it, lots of interesting documents, lots of interesting context.