Jean Valjean does not kill Javert because he knows that whatever Javert has done to him, whatever Javert might wish on him, killing him is the wrong thing to do. Valjean spent years in prison for stealing a loaf of bread, and he spent the years after his imprisonment running from the law—and running from Javert specifically. While killing Javert would end Valjean's years of running and hiding, the murder would hang over him forever.

Valjean is also emulating Bishop Myriel, who could have had Valjean arrested for stealing from him. Though the law technically dictated that Valjean should have been arrested, Myriel chose instead to defend Valjean and give him the silver. This generous act allowed Valjean to change his life and become a good man. When Valjean spares Javert, he is following the example set by the bishop by acting selflessly. While the generous act allowed Valjean to turn his life around, however, it makes Javert question his dedication to the law, which ultimately leads to his suicide.

In not killing Javert, Jean Valjean continues his spiritual transformation.  Since his experience with guilt and regret, Valjean had spiritually evolved into a higher form of being in the world.  This condition had been brought on by a universal understanding of love and compassion.  The same traits that the Bishop showed to Valjean in his redemption are the qualities that he emanates to the world.  These qualities drive him to take care of others such as Cosette and Marius, embrace philanthropy, and seek to make the world better than what it is.  These traits also enable him to refrain from killing Javert. 

Valjean does not kill Javert because of the promise and hope that is within his spirit.  Valjean realizes that if he wishes to break the cycle of being trapped in the moral abyss of the world around him, he has to act in a manner that is transcendent from it.  In this, he does not kill Javert, but rather allows him to be free.  Valjean gives life, as opposed to taking it.  The ability and desire to transform what life should be as opposed to what it is defines Valjean, and is the motivating factor behind why he does not kill Javert at the moment when he could have done so.

In Les Miserables, why didn't Inspector Javert kill or imprison Jean Valjean?

Inspector Javert actually does arrest Jean Valjean once, after Valjean comes forward to save the wrongly accused Champmathieu at the end of Part I, but Valjean manages to escape.  Valjean spends the following years living a peaceful life incognito in Paris, but he must be always watchful so that Javert does not find him.  Decades later, Javert's and Valjean's paths cross again, when Valjean saves Javert's life.  In an act of supreme honor, Valjean, who has Javert at his mercy, reveals his identity and where he can be found, and lets him go.  Javert, impressed by Valjean's courage, begins to question his own beliefs and the single-minded manner in which he has sought over the years to bring Valjean to justice.  Amazed at the integrity of the man he has spent a lifetime pursuing, Javert finds himself in a moral quandary which he cannot resolve, and instead of going after Valjean and completing his life's endeavor, he drowns himself in the river.