In the United States, why aren't both legislative chambers involved in the Supreme Court confirmation process?

Because the Supreme Court was designed to address only constitutional disputes the rulings with which the otherwise sovereign states must comply. The Senate is the chamber representing the states. The House was parochial in nature and a body of direct representatives of the people, most of whom the Founders regarded as rabble.

Given the current oval office occupant, it appears the rabble has had the last laugh.

The existing answers are good, but I think they miss one element of why the Senate performs this function and not the House:

Senators were originally appointed by state governments. The scope of federal law was originally quite slight, and a substantial amount of federal court business was naturally anticipated to deal with disputes between states. Having federal justices appointed with the advice and consent of a body controlled by state governments gave those state governments a check on the power of the federal courts to impose judgments on the states.

Using the House for this function would have given the large states an advantage over the small states. Several elements of the Constitution are designed to prevent such an advantage.

The framers didn't trust democracy or the people (at the federal level)

A shocking sentiment to many, perhaps, but a fact nonetheless. They often saw direct democracy on any given matter as either too onerous a burden on the people (believing they should put in great diligence to exercise the power wisely, but fearing they wouldn't due to lack of time and resources), or that the people as a whole or in groups were too susceptible to various corrupting vices. A fear they were quite proud of, especially the various ways in which they managed to minimize the impact of the people on the federal government. The easiest source of establishing this is to read the Federalist papers, which are the closest thing we have to historical accounts of what exactly the framers were thinking (the actual meetings were kept secret; the Federalist papers are somewhat more intellectual propaganda to encourage passage of the Constitution). For the particular matter at hand, Federalist paper #76, written by (or at least attributed to) Hamilton, seems most relevant.

Hamilton argues why the power of appointment should not be vested in the people or a group of people as so (emphasis mine):

It will be agreed on all hands, that the power of appointment, in ordinary cases, ought to be modified in one of three ways. It ought either to be vested in a single man, or in a SELECT assembly of a moderate number; or in a single man, with the concurrence of such an assembly. The exercise of it by the people at large will be readily admitted to be impracticable; as waiving every other consideration, it would leave them little time to do anything else. When, therefore, mention is made in the subsequent reasonings of an assembly or body of men, what is said must be understood to relate to a select body or assembly, of the description already given. The people collectively, from their number and from their dispersed situation, cannot be regulated in their movements by that systematic spirit of cabal and intrigue, which will be urged as the chief objections to reposing the power in question in a body of men.

So Hamilton argues that direct, democratic appointments are to be discarded because the people cannot be bothered with such burdens (and implicitly would make unwise and impulsive decisions on the matter, not having the time to consider it properly), and that investing it in just a group of people (such as Congress, either one chamber or both) will be subject to the vices of cabals and intrigues. Thus he arrives at the conclusion that the power of appointment must belong to one man, and the constitution naturally provides that one man as the President (as all other branches are invested in groups of people: institutions with many members). He provides various arguments why a single person has greater discernment and focus than a group of other people, even if those other people are individually smarter and more discerning. It's probably worth noting that his fundamental supposition for this is that the office of the Presidency would be naturally filled by "a man of abilities, at least respectable".

He then proceeds to argue why ultimately the power of only nomination was provided to the President, and that another body was required to approve. Emphasis mine in the following:

To what purpose then require the co-operation of the Senate? I answer, that the necessity of their concurrence would have a powerful, though, in general, a silent operation. It would be an excellent check upon a spirit of favoritism in the President, and would tend greatly to prevent the appointment of unfit characters from State prejudice, from family connection, from personal attachment, or from a view to popularity. In addition to this, it would be an efficacious source of stability in the administration.

It will readily be comprehended, that a man who had himself the sole disposition of offices, would be governed much more by his private inclinations and interests, than when he was bound to submit the propriety of his choice to the discussion and determination of a different and independent body, and that body an entier branch of the legislature. The possibility of rejection would be a strong motive to care in proposing. The danger to his own reputation, and, in the case of an elective magistrate, to his political existence, from betraying a spirit of favoritism, or an unbecoming pursuit of popularity, to the observation of a body whose opinion would have great weight in forming that of the public, could not fail to operate as a barrier to the one and to the other. He would be both ashamed and afraid to bring forward, for the most distinguished or lucrative stations, candidates who had no other merit than that of coming from the same State to which he particularly belonged, or of being in some way or other personally allied to him, or of possessing the necessary insignificance and pliancy to render them the obsequious instruments of his pleasure.

As such the combination of President + Senate was meant to derive all the benefits of singular or group investment of the power, with each countering their individual failings.

The question of why the Senate specifically is not directly addressed, but it is implicit in what has already been stated: the people themselves are not to be trusted, and as the House is itself directly beholden to the people then the House is also not to be trusted. In many ways the House exists as a practical necessity: if the people feel completely divested of representation and power they tend to rebel, and start throwing tea into harbors, or worse yet picking up pitchforks and torches and start killing and destroying to get their way. As such a democratically elected and represented body was necessary, but was then avoided or mitigated in manifold ways. In other federalist papers it is argued that the Senate will be naturally composed of morally upstanding, intelligent, long-term view people with an interest towards the nation as a whole, while the House will be filled with people with short-term goals (they have 2-year terms, compared to 6 in the Senate) with views directed towards their constituency only. The Senate was thereby naturally the body to look at for involving in long-term decisions like lifetime appointments, as well as those affecting specifically the federal government itself (such as appointment of Officers).

As a concluding aside, I'll note that such distrust of (direct) democracy is in fact not a singularly peculiarity of the US, and is present in most otherwise-democractic nations. The basic reason is the same one that guided a lot of the construction of the US constitution: smaller, less populous states/regions will feel a heightened sense of powerlessness; bigger and more populous areas will dictate the direction of the nation, even if to the great harm of the smaller ones; and they would as such feel less reason to participate in such a system, and more reason to rebel against once in it. It is often believed that stability of a large, diverse system is better achieved by "degressive proportionality", wherein smaller regions have proportionally greater representation. The European parliament expressly uses such a system; the German Bundesrat has a fixed 3-6 members per region, even if the most populous region has well more than double the people of the smallest; the US has the Electoral college and the Senate; the UK has the House of Lords, and in some sense the Scottish Parliament and the Northern Ireland Assembly; etc. What's peculiar to the US system is that the non-proportional portion of the legislature, the Senate in this case, is at least as powerful as the proportional portion (and in actual practice is more powerful, as demonstrated by all this controversy on SCOTUS appointments). While the House does have special powers particular to it (enumerated in another answer), the Senate also has such special powers, such as "advice and consent" on appointments, and the Senate powers have proven much more potent than the House ones.

So, as William has mentioned, the fact that the Senate's consent is required in court appointments is established in the U. S. Constitution, Article II, section 2:

He [the President] shall have Power, by and with the Advice and Consent of the Senate, to make Treaties, provided two thirds of the Senators present concur; and he shall nominate, and by and with the Advice and Consent of the Senate, shall appoint Ambassadors, other public Ministers and Consuls, Judges of the supreme Court, and all other Officers of the United States, whose Appointments are not herein otherwise provided for, and which shall be established by Law: but the Congress may by Law vest the Appointment of such inferior Officers, as they think proper, in the President alone, in the Courts of Law, or in the Heads of Departments.

For this to change, it would require an amendment to the Constitution (or a Convention of States, but that is far more uncommon. In fact, it has never happened to my knowledge) which the process of doing this is established in Article V (same source as above):

The Congress, whenever two thirds of both Houses shall deem it necessary, shall propose Amendments to this Constitution, or, on the Application of the Legislatures of two thirds of the several States, shall call a Convention for proposing Amendments, which, in either Case, shall be valid to all Intents and Purposes, as Part of this Constitution, when ratified by the Legislatures of three fourths of the several States, or by Conventions in three fourths thereof, as the one or the other Mode of Ratification may be proposed by the Congress; Provided that no Amendment which may be made prior to the Year One thousand eight hundred and eight shall in any Manner affect the first and fourth Clauses in the Ninth Section of the first Article; and that no State, without its Consent, shall be deprived of its equal Suffrage in the Senate.

As you can probably see, it can be difficult to pass an amendment, though not impossible. In this particular case, it would be very hard to change this, since it would require the agreement of the Senate (as "both Houses" refers to both the House of Representatives and the Senate here) to give up power. So, you could say it is a historical artifact that has never changed, and probably won't do so.

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As to why they did this, the Founding Fathers had a habit of putting in, what they called, "checks and balances"--that is, balancing power among the various parts of the government to keep any one person from becoming too powerful. Here, they give the Senate power to consent to appointments, but they also grant the other house powers as well.

Take for instance Article I, section 8 (again, same source as above):

The Congress [that is, the House of Representatives] shall have Power To lay and collect Taxes, Duties, Imposts and Excises, to pay the Debts and provide for the common Defence and general Welfare of the United States; but all Duties, Imposts and Excises shall be uniform throughout the United States;

Section 8 continues to list a bunch of things that only the House of Representatives can do. And if you look, they do the same for the Executive (or President) as well.

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TL;DR: It is somewhat of a historical artifact that was never changed over the years, and probably will never change. The reason this power was given to the Senate was to create "checks and balances," or to balance the governmental power between all forms of government to keep any one person or group of people from becoming too powerful.

This is because of the Appointments Clause of the Constitution:

... and [the President] shall nominate, and by and with the Advice and Consent of the Senate, shall appoint Ambassadors, other public Ministers and Consuls, Judges of the supreme Court, and all other Officers of the United States, whose Appointments are not herein otherwise provided for, and which shall be established by Law: but the Congress may by Law vest the Appointment of such inferior Officers, as they think proper, in the President alone, in the Courts of Law, or in the Heads of Departments.

https://en.wikipedia.org/wiki/Appointments_Clause