TL;DR: The Tempest does not respect Aristotle’s unity of action, but to discover this you need to read Aristotle, and even literary critics sometimes neglect primary sources.
The unity of action comes from the Poetics of Aristotle:
After these definitions we must next discuss the proper arrangement of the incidents since this is the first and most important thing in tragedy. We have laid it down that tragedy is a representation of an action that is whole and complete and of a certain magnitude, since a thing may be a whole and yet have no magnitude. A whole is what has a beginning and middle and end. A beginning is that which is not a necessary consequent of anything else but after which something else exists or happens as a natural result. An end on the contrary is that which is inevitably or, as a rule, the natural result of something else but from which nothing else follows; a middle follows something else and something follows from it. Well constructed plots must not therefore begin and end at random, but must embody the formulae we have stated.
Aristotle (c. 335 BCE). Poetics 1450b. Translated by W. H. Fyfe (1932).
Fyfe’s translation is not very idiomatic (it was probably intended as an aid to construing the original Greek), but we can get some sense out of it. In this passage, Aristotle is in the middle of defining what he means by “tragedy”, and one component of that definition is that a tragedy should consist of a “representation of an action that is whole and complete”. He seems to mean that an action is “whole and complete” if it consists of a chain of causes and effects, starting at the beginning (an event “which is not a necessary consequent”: that is, an accident or coincidence), and going on to the end (an event from which nothing, or at least nothing important to the plot, follows).
This becomes clearer if we look at the kinds of ancient Greek plays that Aristotle is discussing in this passage. In Sophocles’ Oedipus Rex, the event that causes the plot is Oedipus becoming king of Thebes by unknowingly killing his father Laius and marrying his mother Jocasta. Everything else in the play—Jocasta’s suicide, Oedipus’ self-mutilation, his and his daughters’ exile—follows from this original event in an inexorable chain of consequences. But the original event itself has no explicable cause—it is a self-fulfilling prophecy, or a divine curse, or a monstrous coincidence.
Aristotle’s subsequent remarks show that cause and probable effect are the keys to his idea of the unity of action:
To give a simple definition: the magnitude which admits of a change from bad fortune to good or from good fortune to bad, in a sequence of events which follow one another either inevitably or according to probability, that is the proper limit.
[…] Homer, supreme also in all other respects, was apparently well aware of this truth either by instinct or from knowledge of his art. For in writing an Odyssey he did not put in all that ever happened to Odysseus, his being wounded on Parnassus, for instance, or his feigned madness when the host was gathered (these being events neither of which necessarily or probably led to the other), but he constructed his Odyssey round a single action in our sense of the phrase.
Aristotle, Poetics 1451a. My emphasis.
But Aristotle’s idea of “probability” is pretty flexible, as in this example where he allows it to include a “providential” coincidence:
since tragedy represents not only a complete action but also incidents that cause fear and pity, and this happens most of all when the incidents are unexpected and yet one is a consequence of the other. For in that way the incidents will cause more amazement than if they happened mechanically and accidentally, since the most amazing accidental occurrences are those which seem to have been providential, for instance when the statue of Mitys at Argos killed the man who caused Mitys’s death by falling on him at a festival. Such events do not seem to be mere accidents. So such plots as these must necessarily be the best.
Aristotle, Poetics 1452a.
So does The Tempest respect Aristotle’s definition of the unity of action? Clearly the answer is going to depend on how far you are willing to stretch the idea of cause and effect being “natural”, “probable” or “inevitable”. Granted that Prospero has been exiled from Milan to an island in the Bermudas, and granted that he is a magician with powerful spirits at his command, and granted that his enemies happen to be on a ship within his magical reach, then the events of the play do follow as consequences. But these events do not seem to be “probable” or “inevitable” in the sense of Aristotle. And this applies not only to the magical events: is it really probable that Ferdinand and Miranda should fall in love at first sight and consent to be wedded within the hour? Is it really inevitable that Alonso, who conspired with Antonio against Prospero, should sincerely repent and beg forgiveness? Perhaps some of these can be filed under the heading of providential coincidences, like the statue of Mitys, but to do this for all of the fantastic events of the play seems to stretch things too far.
What about subplots? Aristotle’s criterion for the unity of action is that all the “component incidents” of a plot must contribute to the chain of cause and effect, and that if an incident can be removed without materially changing the rest of the plot then this spoils the unity:
As then in the other arts of representation a single representation means a representation of a single object, so too the plot being a representation of a piece of action must represent a single piece of action and the whole of it; and the component incidents must be so arranged that if one of them be transposed or removed, the unity of the whole is dislocated and destroyed. For if the presence or absence of a thing makes no visible difference, then it is not an integral part of the whole.
Aristotle, Poetics 1451a.
The Caliban subplot in The Tempest is clearly not “integral” in this sense. It is foiled so easily by Prospero and Ariel that it makes no difference to the events or outcome of the main plot.
The Tempest fails to respect Aristotle’s unity of action, both in the improbability of the events in the main plot, and in the failure of the subplots to make a difference to the outcome of the main plot. So what makes so many critics claim otherwise? This is hard to answer without going into unverifiable speculation, but I have three theories.
First, there is always a temptation to distort the facts in order make a more interesting claim. Aristotle’s unity of action is pretty flexible, so that if you want to claim that The Tempest respects it, you can always justify yourself with something like, “if the statue of Mitys counted as ‘probable’ enough for Aristotle’s purposes, then why not a victimless shipwreck brought about by a magical spirit?”
Second, some critics may not have looked carefully at the text of Aristotle’s Poetics, but have been working from secondary or tertiary sources that summarize or explicate Aristotle, omitting or generalizing some of the detail in the original. You can see that something like this must have been happening because Pickering’s criterion for subplots is that they must “belong organically” to “a single main plot”, which does not seem to correspond closely to anything in Aristotle. Similarly, the CliffsNotes series is intended for people who don’t want to read original texts, so it would not be so surprising if its contributors didn’t always read original texts either.
Third, some critics may not even have looked at summaries of Aristotle, but have simply copied the claim that The Tempest respects the unity of action from a secondary or tertiary source. Once an interesting factoid like this becomes established, it is almost impossible to correct.