How does Hamlet fall prey to inaction?

Aristotle teaches that every tragic hero not only has honor but also has a tragic flaw. As the tragic hero of Hamlet by William Shakespeare, what is Hamlet’s tragic flaw? There are four possible tragic flaws often debated by scholars: inaction, melancholy, trust in the ghost, or madness. Every single one of these tragic flaws can be proven using the text. Likewise, every single one of them can be disproven. (Further, and probably much to your chagrin, there are other scholars, such as Bradley for example, who are adamant that Hamlet doesn’t have a tragic flaw.) Still, let’s explore each of them in turn so that, as readers of Hamlet and as examiners of its titular character, we can decide which one of Hamlet’s tragic flaws we can sink our teeth into.
Perhaps the most common tragic flaw pointed to by scholars (and most notably by Samuel Taylor Coleridge) is the tragic flaw of “inaction.” Even if a particular scholar doesn’t think that Hamlet’s flaw is inaction, that scholar must at least admit that inaction is a prominent theme in Shakespeare’s Hamlet.
As readers, we come upon the first example of inaction in Act I. Hamlet has just learned of his father’s murder by his uncle, Claudius, but instead of avenging his father, Hamlet decides to do a kind of “test” on each of the characters to determine their knowledge. How does he perform this test? By putting an “antic disposition on,” in other words, by acting crazy. How much of this is Hamlet’s noble character and how much of this is procrastination? It is up to the reader (and in some cases, the director) to decide.
Hamlet comments on his possible tragic flaw first by saying, "The time is out of joint. O cursed spite, / That ever I was born to set it right." In other words, Hamlet doesn’t really want to avenge his father at all. Time isn’t even on his side. He doesn’t feel responsible for setting the divine order to rights again. To do so would be “spite” according to Hamlet. Such are the various and sundry reasons for his first delay.
Secondly, most scholars who take Hamlet’s flaw to be inaction point to Hamlet actually naming his own tragic flaw found within one of his monologues. In very few words, Hamlet states, "I am pigeon-livered and lack gall." Not only does Hamlet name and own his own flaw here, but he is unable to correct it. As evidence, even after concocting a very ingenious plan to catch his uncle in the act (by commissioning the players to perform the play within a play called The Murder of Gonzago, which relates the exact replication of the murder of Hamlet’s father), Hamlet simply becomes upset with his own inactive self and gives the flaw a name in the line above.
Third, the reader should look at Hamlet’s famous “to be or not to be” speech in act three. Most scholars agree (although there would be some that would argue with me) that this particular monologue is Hamlet’s thoughts to himself about merely existing. In other words, it is Hamlet contemplating suicide. If Hamlet simply contemplates suicide, without carrying it out, it is another example of inaction. (It is probably important to mention here that the scholars who do not think this speech is about contemplation of suicide agree that it is at least about the troubles of turning thought into action and, as such, can still be an example of Hamlet’s inaction.) Later in that soliloquy, Hamlet says the following: “Thus the native hue of resolution / Is sicklied o'er with the pale cast of thought, / And enterprises of great pitch and moment, / With this regard their currents turn awry, / And lose the name of action." Note the perfection of the words “lose the name of inaction.” Hamlet, again an example of inaction within his own words. Hamlet, then, is not able to avenge his father.
As a final example, it is important to look at the scene where Hamlet comes upon Claudius praying for forgiveness. Let’s look at the important lines spoken by Hamlet in order to examine them further: “Now might I do it pat, now “a is a-praying, / And now I’ll do’t. And so goes to heaven, / And so am I revenged. That would be scanned. / A villain kills my father, and for that / I, his sole son, do the same villain send / To heaven.” Again, Hamlet doesn’t act. It is a perfect scene, in private, where the avenging can be done without disturbing anyone else. Why doesn’t Hamlet act? (In reality, the reasoning is unimportant; however, for the sake of argument, let’s speak of it anyway.) Hamlet doesn’t act because he will send Claudius’s soul to heaven if he is in the true state of contrition when he is killed. Hamlet doesn’t want to send his father’s murderer to heaven. Hamlet literally wants Claudius to go to hell. Therefore, because of that desire, Hamlet again doesn’t avenge his father.