KING HENRY Tomorrow toward London back again, To look into this business thoroughly, And call these foul offenders to their answers, And poise the cause in justice' equal scales, Whose beam stands sure, whose rightful cause prevails. (2.1.214-219)
Hearing that Eleanor has been arrested for witchcraft, Henry comforts himself with the idea that justice will prevail. Perhaps he's just too naive to see the scheming that's going on all around him, or maybe he really believes in justice. Either way, he seems to want a system that he can rely on.
GLOUCESTER Eleanor, the law, thou seest, hath judged thee. I cannot justify whom the law condemns. (2.3.17-18)
Gloucester is sad—crying, even—about his wife's sentence, but he openly says that what she did was wrong. He's a man of the law, and he follows it through and through, even when it's difficult. Henry wants to believe in a system, but comes across as naive for doing so; Gloucester, on the other hand, seems strong and principled when he holds his wife to such high standards. Henry seems to want to shirk responsibility, while Gloucester takes it on, even when it's hard.
GLOUCESTER So long as I am loyal, true and crimeless. Wouldst have me rescue thee from this reproach? Why, yet thy scandal were not wiped away, But I in danger for the breach of law. (2.4.64-67)
When Gloucester hears Eleanor talking about how hard it is to walk in the streets displaying her criminal status, he separates himself from her. Gloucester feels for his wife, but he cares more about justice being served. For him, that means he must be above reproach when it comes to the law. Too bad it doesn't do him any good.
GLOUCESTER Well, Suffolk, thou shalt not see me blush Nor change my countenance for this arrest. A heart unspotted is not easily daunted. The purest spring is not so free from mud As I am clear from treason to my sovereign. Who can accuse me? Wherein am I guilty? (3.1.99-104)
Gloucester's arrested for treason, but he tells Suffolk he's not worried, because he's done nothing wrong. We know it doesn't matter whether Gloucester is guilty or not, but we think it matters that he's so invested in the system of justice. He doesn't even fight when he's being taken down, because he's confident that he will be judged fairly. It's an interesting twist of events: do you think Eleanor was judged fairly? Maybe she really did commit a crime, but should it have been considered a crime to begin with? Do Gloucester and Eleanor share a similar fate? Would he have been able to escape if he hadn't had such faith in the system?
QUEEN MARGARET Believe me, lords, were none more wise than I— And yet herein I judge mine own wit good— This Gloucester should be quickly rid the world, To rid us of the fear we have of him. (3.1.233-236)
Margaret doesn't care one iota about justice—she wants Gloucester dead regardless of what he's done, because he's in the way. Like her pals Suffolk and Beaufort, Margaret doesn't care about integrity. Margaret and company decide to make the law and conform it to their own will. Their actions take justice to a new level, and some of them pay for what they did.
KING HENRY O Thou that judgest all things, stay my thoughts, My thoughts, that labor to persuade my soul Some violent hands were laid on Humphrey's life. If my suspect be false, forgive me, God, For judgment only doth belong to Thee. (3.2.140-144)
As Henry prays to find out what happened to Gloucester, he tells us a lot about his own personal form of justice. Unlike his wife or his nobles, Henry doesn't take matters into his own hands. He knows something's wrong with Gloucester's death, but he lets God take care of it. Henry's conception of justice is divine and omniscient; it's up to God to decide who is guilty and who is not. As far as that goes, he's following Christian teaching pretty closely, but it's not totally clear if he's doing so out of Christian charity or if he's primarily looking for an excuse not to get his hands dirty.
KING HENRY What stronger breastplate than a heart untainted? Thrice is he armed that hath his quarrel just, And he but naked, though locked up in steel, Whose conscience with injustice is corrupted. (3.2.240-243)
At this point, Henry doesn't seem to have any idea that his wife is a lying murderer. He's most concerned with the fact that Gloucester seems to have died unjustly. It's a revealing comment: Henry actually thinks that justice exists in his kingdom, as is. In reality, there's injustice going on all around.
KING HENRY Go, take hence that traitor from our sight; For his death we do perceive his guilt. And God in justice hath revealed to us The truth and innocence of this poor fellow, Which he had thought to have murdered wrongfully. (3.3.102-107)
Everyone is surprised when Peter busts it out and wins the swordfight against Horner. He had no training and no skills, so Henry figures that this must be a sign that God had judged Peter innocent (and that it has nothing to do with the fact that Horner was tanked). He stays out of it completely and lets God take the wheel.
SAYE Justice with favor have I always done; Prayers and tears have moved me; gifts could never. When have I aught exacted at your hands But to maintain the king, the realm, and you? Large gifts have I bestowed on learnèd clerks, Because my book preferred me to the king, And seeing ignorance is the curse of God, Knowledge the wing wherewith we fly to heaven, Unless you be possessed with devilish spirits, You cannot but forbear to murder me. (4.7.67-76)
Even though Cade and company are about to kill him, Lord Saye remains convinced that they cannot, and will not, simply because he's done nothing wrong. Hmm… where have we heard that before? Oh, yeah: from Gloucester, right before he was murdered. Saye believes in justice, but he dies, anyway.
YORK So let it help me now against thy sword As I in justice and true right express it! (5.2.24-25)
In the beginning of the play, the fight was over who was controlling the king, but by this point, York is ready to take the crown all for himself, without Henry in the picture at all. The thing is, Shakespeare never comes out and says he's right or wrong; he just throws in the word "justice" and lets us decide.