Dr. Chillingworth was what we could psychologically describe as a "passive-aggressive" man, and what we can socially describe as a man "with not enough gall" to face the music.
In being "passive-aggressive" means that he would operate his plans using all his enemies while pretending to be their friend or caregiver (he did this with Dimmensdale). In being a man "with not enough gall" means that he never stood up for himself the way he should have, so, he opted for the cheap and easy way to see Hester and later onnDimmensdale suffer and enjoying it from a distance.
In not so many words: Chillingworth is not man enough to say who he is, and what he has gone through publically. Surely this could have shamed him, but he could have gained much more from people's support than to sit in the back and watch the theatricals without actively taking part of it and see his enemy go down.
(I just really think Chillingworth was never man enough and maybe that's why Hester cheated on him)
In this chapter, Hester Prynne is up on the scaffold to endure her three hours up there in front of everyone. A crowd has gathered to watch her.
In the crowd is the man she is married to but who has disappeared for years. He is calling himself Roger Chillingworth. I would argue that he does not reveal himself because he wants to take revenge. He feels anger toward whoever has been sleeping with his wife and he wants to get back at them. If he revealed himself, he would not be able to achieve this.
It is, indeed, helpful to the reader of Nathaniel Hawthorne's The Scarlet Letter to always consider the context of the narrative as built around the strict Puritanical code of the Massachusetts colony. In the previous chapter, for instance, the cold, smug glee with which the goodwives condemn Hester and suggest more violent torture dramatizes the "fire and brimstone" tactics of early American Puritanism.
Because even the tiniest infraction of Puritan law was met with severe punishment--a fact that Hawthorne found objectionable in early Puritanism--Roger Chillingworth does not reveal himself. For, if he were to come forth and admit to being Hester's husband, he himself would be interrogated by the Puritan leaders and possibly punished for abandoning his wife, living with the Indians, and anything else that could be held against him. Furthermore, he does not wish to draw attention to himself as he plans a plot against the man who has made him a cuckold.
The fact that he feels some guilt about Hester's being abandoned is evident in Chapter IV when he visits Hester, telling her, "We have wronged each other." When Hester asks him, "Why not announce thyself openly, and cast me off at once?" Chillingworth replies,
It may be...because I will not encounter the dishonour that besmirches the husband of a faithless woman.
Then, Chillingworth tells Hester that he seeks no vengeance against Hester, but
the man lives who has wronged us both!....Let him hide himself in outward honour, if he may! Not the less he shall be mine!
Chillingsworth has been away for a considerable amount of time. He had been captured by Indians and taken hostage until he was able to get away. He was probably somewhat disorientated to what was going on. Once he found out, he was thinking things through.
I think he had not had time to think about what his next step should be. He had been gone from the situation so everything must have been new and surprising to him. No one knows for sure what he was thinking but I am sure he was just as curious at the moment as to who was involved with his wife. He was probably thinking how he could find out the truth and dwelling on that instead of revealing his identity.