Judge Irwin is Jack's real father, and he shoots himself in the heart rather than reveal this secret. He's proud. He's already waited way too long to tell Jack, thinking that acting like a father would be enough. In his mind the secret (mostly because it's been a secret for so long) has taken on so much weight that he doesn't know what to do with it. That Ellis Burden and Judge Irwin were childhood friends (much like Adam and Jack), and that betrayal in some ways destroyed Ellis, are no small part of the shame wrapped up in Judge Irwin's true identity. Most of all, the Judge seems to fear Jack – how Jack will judge him. For each day that the truth remains a secret, a bit of weight is added to it. The burden becomes harder to bear.
It's not simply Jack's bringing up the Mortimer Littlepaugh-Governor Stanton business of long ago that drives Judge Irwin to suicide. It's also that Irwin knows that if Jack were aware that he was his father, Jack probably wouldn't be blackmailing him. Jack might well have found out about the Judge's checkered past, but he probably wouldn't have used that information for political purposes. Furthermore, now the Judge can't tell Jack that he's his father without looking like he's doing it to somehow get out of the blackmail situation.
Judge Irwin is probably proud of Jack as well, in a grim way. It was, after all, Judge Irwin who sparked Jack's interest in history. Like Jack, Irwin is an expert historian, and his specialty is the history of warfare and weaponry. But finding the dirt on "the Upright Judge" is a real pièce de résistance. It's a seven-month "journey into the past" that positively blows the lid off of everything and sparks a chain of tragedies.
Yet, from this tragedy, Jack, Anne, and Mrs. Murrell are able to find a sort of peace with the world and their pasts. Judge Irwin meant his death as a gift. The bullet through the heart makes that clear. That's the tragedy of Judge Irwin. With all the grim secrets of the South in those days (as we see in the Cass Burden story), it's no wonder that Judge Irwin lets a bullet to the heart do the talking for him. Slavery can even be thought of as a bullet through the collective heart of humanity.