Ambrose Bierce

The note that Jimmy leaves with the plainclothes detective to give to Bob should prove that Jimmy had only slight misgivings about having Bob arrested. 

Bob: I was at the appointed place on time. When you struck the match to light your cigar I saw it was the face of the man wanted in Chicago. Somehow I couldn't do it myself, so I went around and got a plainclothesman to do the job.
                                                                             JIMMY.

O. Henry didn't want to have Jimmy arrest Bob because that would have required a big scene in which Bob appealed to their old friendship and Jimmy had to explain that he had a duty to perform. The truth is that the friendship ceased to exist the moment Bob lighted his cigar and Jimmy realized who his old friend had become over the past twenty years. Bob might have been shocked at that point, too, if he could have seen Jimmy face and his uniform; but he couldn't see Jimmy because he was blinded by the light of his own match. He only saw the vague figure of a uniformed police officer. If he could have seen Jimmy at that point, Bob might have realized that their old friendship had gone out just like his match. He couldn't be friends with a cop, and Jimmy couldn't be friends with a crook. People change over the years, and friendships burn out just because people change.

There is no question that Jimmy is justified in what he does to his friend. He should do it even if they were still good friends, but they are not really friends any longer. Bob was mistaken to think that he could spend twenty years leading a life of crime and then go back and expect Jimmy to have the same high regard for him as before. Jimmy was basically an honest, responsible man, and he wouldn't have liked Bob after twenty years even if he hadn't become a cop. O. Henry was making the contrast between the two men extreme by having Jimmy turn into a cop, but the relationship would have been over even if Jimmy had become an office worker or something else. 

It is very unwise to make friends with dishonest people, or to maintain friendships with people who turn out to be dishonest. It is best to terminate such friendships completely, regardless of hurt feelings. Such "friends" only cause you trouble, just as they inevitably cause themselves trouble. Jimmy's note not not only explains how he recognized Bob and why he didn't arrest him himself, but by implication it is saying a final and irrevocable goodbye.

People change a lot in twenty years. In O. Henry's story the reader has a sort of "double exposure" perception of both Jimmy and Bob. The reader can visualize them as they looked when they were young and at the same time visualize them as they have matured and as they have changed through their respective experiences over two decades. When 'Silky' Bob lights his cigar in the doorway of the hardware store, Jimmy is probably shocked to see how much his old friend has changed.

The man in the doorway struck a match and lit his cigar. The light showed a pale, square-jawed face with keen eyes, and a little white scar near his right eyebrow. His scarf pin was a large diamond, oddly set.

No doubt if Jimmy were looking at the same man he had said goodbye to twenty years earlier, he might have had serious misgivings about arresting him, or having him arrested by proxy. But this is not the same man. This is a virtual stranger. He has been changed by the same physiological processes that change all men by the time they reach middle age; but he has also been changed by the kind of life he has been leading. It is written all over him. Instead of looking like an ambitious young man starting out in life, he looks like what he is--a hardened criminal. Maybe Bob used to be Jimmy's friend, but he is not Jimmy's friend anymore. And therefore Jimmy has no compunctions about having him arrested. Jimmy and Bob are no longer friends. Friendships rarely last in this world. Jimmy wouldn't want to be associated with a man like 'Silky' Bob. His old boyhood friend has ceased to exist, just as 'Big Joe' Brady's restaurant has ceased to exist.

“It [the passage of twenty years] sometimes changes a good man into a bad one,” said the tall man. "You've been under arrest for ten minutes, ‘Silky’ Bob. 

Hello! 'After Twenty Years' is a short story with an interesting twist. In the story, Bob is waiting for his friend, Jimmy Wells. They have not seen each other for twenty years. Years ago, both agreed that twenty years would be enough time to have their individual destiny worked out and their fortunes made. When a policeman on his night duty walks up and talks to Bob, Bob tells him about his friendship with Jimmy Wells and how he is looking forward to seeing Jimmy after twenty years. Bob tells the policeman that  he's been out west and that 'it takes the West to put a razor edge' on a man. The policeman soon walks away and another man walks up. Bob greets him as Jimmy, but later exclaims

"You're not Jimmy Wells...Twenty years is a long time, but not long enough to change a man's nose from a Roman to a pug."

It turns out that Jimmy did indeed come by: as the policeman earlier. Jimmy Wells sends a plain-clothes officer to arrest 'Silky' Bob because he recognized him as a wanted criminal in Chicago. In Jimmy's note, he tells Bob that he couldn't bear to arrest his friend himself, and that is why he sends someone else to do the job.

So, the question is: is Jimmy's act justified? I imagine you will find those who say it is and those who disagree. To help you make a decision: consider putting yourself in Jimmy's place. Could you arrest someone who you once thought of as your friend? Imagine your shock when you recognize that this good friend with whom you once had great times with, is now a wanted criminal. In the space of a few moments, your joy at seeing him after twenty years is turned to grief. You are torn between your duty as a police officer and your personal feelings for your friend. Jimmy's sincerity as a law man is evident. If you let on that you recognize your friend as a wanted criminal, he might beg you to let him go, on account of your friendship. How would you deal with such a proposition? Jimmy has an interesting solution.

As a friend, Jimmy chooses not to confront Bob, preferring to leave intact the honor and memory of their past friendship; as a police officer sworn to defend justice and protect society, he has no choice but to make sure Bob is apprehended, and he fulfills his duty admirably. I think Jimmy stays true to both his humanity and to his duty.

O' Henry is famous for his open-ended short stories. One imagines that he respects his readers enough to trust that they are able to come to the right conclusions after they read his stories.

Hope I have given you some food for thought for answering this interesting question!