Why does Proctor confess but refuse to name others or let Danforth keep his signed paper in Act 4 of The Crucible?

In act 4, John Proctor reluctantly confesses for two significant reasons. The first reason Proctor confesses is to save his own life so that he can provide for his family. The second reason he decides to confess is because he doesn't feel righteous enough to die as a martyr like Rebecca Nurse or Martha Corey. Proctor tells his wife,

"I cannot mount the gibbet like a saint. It is a fraud. I am not that man." (Miller, 137)

When Danforth begins to question Proctor about whether other citizens were involved in witchcraft, Proctor refuses to accuse them because he knows they are innocent. Proctor is aware that the court is corrupt and refuses to accuse innocent Christians, as this will only support the same court he is challenging. Proctor does not want to have Rebecca or Martha's blood on his hands and already feels overwhelmed with guilt. He is also friends with Rebecca Nurse and Martha Corey and refuses to falsely accuse them. He tells Danforth,

"I have three children - how may I teach them to walk like men in the world, and I sold my friends?" (143)

John Proctor refuses to sign his name to his confession because he would be publicly tarnishing his reputation. In the community of Salem, a person's reputation is significant, and Proctor tells Danforth,

"Because it is my name! Because I cannot have another in my life! Because I lie and sign myself to lies! Because I am not worth the dust on the feet of them that hang! How may I live without my name? I have given you my soul; leave me my name!" (143)

In Act 4 of The Crucible, why doesn't Proctor "name names"?

It at this particular point in the play where John Proctor ascends into a transcendent realm of great literary characters.  It is in this act, in particular the last scene, where the importance of "name" is important, proving to be one of the lasting legacies of the drama.  Proctor has admitted, albeit in a lie, to seeing witches and being guilty of witchcraft.  Rebecca Nurse is brought in to see this, in the hopes she will follow suit.  It is at this moment when John recognizes his own mistakes and simultaneously his own capacity for moral greatness.  Rebecca is "astonished" that John would do such a thing and John catches this glimpse of himself in her own eyes.  It is through this that when John is forced to "name names," he refuses to do so and recants his confession as a lie, guaranteeing his own death.  He cannot name names because it would convict innocent people as well as cause harm to those who remained true to their own name.  John's refusal to name names is actually done to protect others' names.  John is not one to make himself out to be a martyr, but rather he recognizes the need to "make things right" in a setting where so much is wrong.  Someone has to break the moral corruption and hypocrisy that has taken a hold of Salem.  In the first scene of the act, there was ample discussion of how citizens banded together in Andover when a similar predicament plagued their community.  This might not be as evident in Salem, but John's stance is a hopeful start at redemption.  John recognizes this, which is why Elizabeth ends the drama with her own statement:  “He have his goodness now. God forbid I take it from him!”  In a context where there is so little, "goodness," John's refusal to "name names" in recognition of the importance of name is a starting point towards redemption.

In The Crucible, why does Proctor confess, refuse to name names, and withhold his signed paper?

In the last act of the play, John Proctor is facing death by hanging for the crime of witchcraft. When he initially confesses, it is to save his life so that he will be there for his children. Elizabeth, you'll remember, is also sentenced to die after she gives birth to the baby she is carrying. 

Proctor's confession would let him live and his children would have at least one parent left to them. 

An addition reason for Proctor's willingness to confess comes from the fact that he has been judged only on lies. Abigail and Mary Warren have hung the mantel of witchcraft on Proctor without any facts or evidence to back up their claims. Proctor is simply not guilty.

He has already confronted the court and exposed the lies at the root of the Salem witch trials, yet the court persists in carrying out its death sentences. Danforth's reasons for continuing are far from morally convincing. In Proctor's eyes, the court has no moral standing whatsoever. Lying to them to save his life is, for this reason, not really a lie. He is merely feeding the court according to its nature. 

Proctor stops short of naming names of others because he does not want to condemn anyone. He only wants to save himself. He feels that he is weak enough to allow himself to confess to something he did not do, but he is not so weak as to bring others down with him. He has too much regard for the worth and value of others to condemn them. 

He cannot bear the knowledge that his signature will be used to condemn other innocent citizens.

Finally, Proctor finds that he has too much regard for his own worth and his own integrity to sign a written confession. Lying on record is more of an actual lie than making a false statement to a group of people who cannot hear the truth anyway. Lying on record is a step too far for Proctor. His pride and integrity will not allow it. 

The anguish that stops him short is expressed in his climactic lines: 

Because I lie and sign myself to lies! Because I am not worth the dust on the feet of them that hang! How may I live without my name? I have given you my soul, leave me my name!"