in the presence of Abigail.

Proctor is uncomfortable with Abigail's confiding in him on the nights events. He and Abigail had engaged in an illicit affair, and they had been caught by Elizabeth, his wife. This is why she does not trust her husband. Their marriage is in a constant state of turmoil because of it. Proctor does not go to tell what he knows because he feels that he cannot prove that Abigail said it if she decides to deny it. He also does not put a lot of stock into the powers that be.

Furthermore, he does not want to have to discuss why Abigail would be confiding in him in the first place. He is torn up with guilt, and does not wish to cause Elizabeth any more pain or embarrassment.

Elizabeth wants him to go and report what he knows because it is the right thing to do. Elizabeth is a god-fearing woman who believes in telling the truth. The situation is getting completely out of hand, and she feels that this would help.

John Proctor is hesitant to reveal the fact that Abigail has told him she and the other girls were just "sporting" in the woods mainly because this would indicate to his wife Elizabeth that he has had one-on-one contact with Abigail, his former "mistress".  He had previously indicated to Elizabeth that he had only met in a group with Abigail, and this lie would arouse Elizabeth's suspicions.

Proctor does not want to reveal what he knows because of the way he came about the knowledge of Abigail's lies.  He had been having an affair with her, and when he tells her it is over, Abigail promises to ruin his relationship with his wife Elizabeth if he follows through. 

When Proctor tells her, "Abby, I never gave you hope to wait for me," she quickly and angrily responds:  "I've got something better than hope."  Proctor, perhaps for the first time, understands the depth of her desperation and obsession with him.  He is shocked by her hatred of Elizabeth; her accusations are completely untrue and dillusional ("She is blackening my name in the village!  She is telling lies about me!  She is a cold, sniveling woman and you bend to her!")

John knows that if he leaves, she will do to Elizabeth what she has done to so many other innocent women:  accuse her of witchcraft and seal her death warrant.  Abby's obsession with John knows no bounds.  Finally, he gets it. 

On some level, Proctor is a bit leery of going after Abigail's hypocrisy because of the collateral damage on a social level that is going to be caused.  I think that this is what motivates him to try to skirt around or dance around the issues of her motivation, and their relationship.  At this level, I think this is when we see Proctor at not necessarily at his strongest.  Yet, Proctor realizes the folly of this throughout the course of the play when he sees that the failure to stand up to Abigail's accusations ends up ripping apart the social fabric of the town and causes irreparable damage to people's lives and the way in which the town functions.  It is at this moment when Proctor assumes a superhuman status from being quite normal and ordinary in his commitment to speaking the truth, regardless of the cost.  It is something that ends up costing him his life, but is seen as a price worth paying if it will stop the spread of evil.  This level of moral ascendancy is not present at the start of the drama, and could be a major reason why we see Proctor as displaying ambivalence towards Abigail's lies.

John Proctor is having quite the dilemma. Whether in Act I or Act III, Proctor is reluctant to expose Abigail because if he does, she has quite a bit that she can say about him. You know how kids are, if one tries to tell on another, the second child will tell some other detail on the first. This is the same situation. If John reveals Abby's lies, Abby can reveal a secret that is much more shameful in a Puritan society: their adulterous affair. Should Abby bring this detail to the surface, not only would John Proctor's reputation be stained, but under the current circumstances he would likely be named as one who was working with the Devil.

Early in The Crucible John Proctor is contrite with his wife Elizabeth and seeks her forgiveness for the affair he had with Abigail Williams. 

At the beginning of the play, Proctor attempts to convince Abigail that their affair is completely over and that she should expect nothing more from him. During this conversation, Abigail tells Proctor why Betty is in bed, suggesting that Betty is not really sick at all. 

She tells him that it is all pretense and that Betty is just scared. 

 This knowledge becomes important when accusations of witchcraft begin to be made against townspeople. Proctor tells Elizabeth what Abigail had told him and Elizabeth insists that Proctor share his knowledge with the authorities before things could get out of hand. 

Proctor believes that the truth will soon enough come out on its own, but Elizabeth insists that he go to Salem. When he refuses Elizabeth believes that he is trying to protect Abigail. Her suspicions are an outgrowth of her jealousy and pain at having been cheated on by John Proctor. She is not confident that the affair is completely over, despite Proctor's repeated claims that it is now ended. 

Just to add to the answer above...Proctor tells Elizabeth that he wonders if anyone would believe him if he told - he says he has no way to prove it because Abigail told him that when they were alone.  This, of course, upsets Elizabeth again because Proctor had originally said there were othe people around when he had seen Abigail.