I thought I had read that he fired off - as was often the case in duels, but that Burr did not, which is why many even on his side were upset at him....gotta go dig.
ah...a matter of contention.
Quickly from Wikipedia:
“David O. Stewart, in his biography of Burr, American Emperor, notes that the reports of Hamilton’s intentionally missing Burr with his shot only began to appear in newspaper reports published in papers friendly to Hamilton in the days after his death. However Ron Chernow, in his biography, “Alexander Hamilton”, states Hamilton told numerous friends of his intention to avoid firing at Burr well before the duel.[13] Additionally, according to Chernow, Hamilton wrote a number of letters dated before the duel that also attest to this intention. The two shots, witnesses reported, followed one another in close succession, and none of those witnesses could agree as to who fired first.”