Why place a hand on the Bible instead of the Judge's genitals when taking an oath?

Testo an Italian word derives from the Latin term, textum (text), which in turn originated from the verb texĕre, which means to weave.

Treccani, the Italian online dictionary and encyclopedia par excellence says:

testo² s. m. [dal lat. textum -i o textus -us, rispettivam. part. pass. neutro e der. di texĕre "tessere"]. [...] 2. (estens., bibl.) [edizione, spec. se antica e autorevole: t. classici; i t. sacri] ≈ libro, opera, scritto, volume.

Roughly translated, the word testo is used when referring to the classics i.e. classic literature, and sacred works or volumes.

The verb, testify, (In Italian testimoniare) derives from the noun, testimony, its Latin form testimonium which the OP rightly affirmed derives from testis and its plural form teste.

The testicle (from Latin testiculus, diminutive of testis, meaning "witness" of virility, plural testes)

From Dictionary.com
testis

(pl. testes), 1704, from L. testis "testicle," usually regarded as a special application of testis "witness" (see testament), presumably because it "bears witness" to virility (cf. Gk. parastates, lit. "one that stands by;" and Fr. slang témoins, lit. "witnesses").

Looking at the word teste, which means also witness in Italian, according to Treccani it derived originally from the Latin term, tristis (sad, sorrowful, disagreeable or foul smelling) which later evolved into terstis, meaning third party.

So far I haven't find any solid evidence that suggests testament, testimony, testify, are directly related to testes/testicles. I'm more inclined to believe that the root word is textum meaning text or teste (witness), therefore the tradition of swearing an oath by placing one's hand on the Bible rather than on one's testicles, makes sense. If this needs reminding, Christians are not required to be circumcised, so there is no guarantee the judge would be circumcised. See David M's answer as to why it is relevant.

One more thing to consider, would women really have been asked to place their hands on the judge's genitals before a court of law? Back when?

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From the Wikipedia article entitled Sexuality in Ancient Rome

The apparent connection between Latin testes, "testicles," and testis, plural testes, "witness" (the origin of English "testify" and "testimony")[185] may lie in archaic ritual. Some ancient Mediterranean cultures swore binding oaths upon the male genitalia, symbolizing that "the bearing of false witness brings a curse upon not only oneself, but one's house and future line."[186] Latin writers make frequent puns and jokes based on the two meanings of testis:[187] it took balls to become a legally functioning male citizen. The English word "testicle" derives from the diminutive testiculum.

link 186 informs

Joshua T. Katz, "Testimonia Ritus Italicus: Male Genitalia, Solemn Declarations, and a New Latin Sound Law," Harvard Studies in Classical Philology 98 (1998) 183–217 (quotation from p. 193), pointing to the oaths in the Book of Genesis, chapters 24 and 47; the testicles of ritually slaughtered animals used to affirm testimony in Athenian murder trials, as at Demosthenes, Contra Aristocratem 23.67f.; Rhetorica ad Herennium 3.33, where ram's testicles are a mnemonic device in a courtroom exercise. Katz proposes that the Umbrian hapax urfeta means "testicles" and is related to Latin orbis (as "balls"); thus the Iguvine Tables also make a connection between testicles and "solemn declarations" (Katz, p. 191).

I had to look up hapax because I don't think I have ever come across this word before. A hapax legomenon is a word that occurs only once within a context, either in the written record of an entire language, in the works of an author, or in a single text.

A Slate article, Where Did We Get Our Oath? debunks the urban legend, as it claims, of ancient Romans who vowed to tell the truth by grabbing hold of their testicles or as I found in one recount; of two male witnesses who would hold each other's testicles when taking an oath^{Games Primates Play}, I'm inclined to think this version is rather fanciful and at this point we really need an expert in Ancient History.

Latin scholars have debunked this colorful claim, pointing out that testis more likely comes from the Ancient Greek for "three"—a witness being a third observer of events.

Witness (Testimonio ¹/Testimone ²/Teste ³)

Until the 16th century the Italian for witness used to be testimonio but today that has been superceded by testimone and teste. A brief history of the word, testimonianza, written in Italian, is provided by Treccani. Testimonianza in English is deposition (Law) Sworn testimony recorded for use in court at a later date.

In his book, AIDS, Bearing Witness, and the Queer Moving Image, Roger Hallas provides this clear explanation.

Consequently, it appears that the word, testify, and all its derivatives come from the Latin term, terstis (third party or person) which Treccani also confirms.

I think this is a chicken and the egg argument.

The notion that testimony springs forth from the testicles and/or their use as an agent of swearing truth appears spurious. And, in fact, conceptually the etymology seems to have gone in the opposite direction.

The concept of testicles from the latin testis seems to come from the gonads being a witness to male virility.

In addition to the etymonline entry above claiming it groundless, there are multiple online debunks of the practice of swearing on one's testicles being the origin of testify and related words. Source; Source; Source; Source - There is a fair sampling of what I've found. Most of the "sources" linking the origin to the dubious Roman practice do not provide any etymologic sources.

So, it would appear that the shared root of testis, testimony, testify, etc. is based upon the latin meaning to bear witness, and not to swear upon one's testicles.

For us Jews, circumcision represents an act which fulfills the covenant with God that was made by Abraham.

God said to Abraham, "As for you, you shall keep my covenant, you and your offspring after you throughout their generations. This is my covenant, which you shall keep, between me and you and your offspring after you: Every male among you shall be circumcised. You shall circumcise the flesh of your foreskins, and it shall be a sign of the covenant between me and you. Throughout your generations every male among you shall be circumcised when he is eight days old, including the slave born in your house and the one bought with your money from any foreigner who is not of your offspring. Both the slave born in your house and the one bought with your money must be circumcised. So shall my covenant be in your flesh an everlasting covenant. Any uncircumcised male who is not circumcised in the flesh of his foreskin shall be cut off from his people; he has broken my covenant."

There is generally no mention of the testicles in that act. The testicles have no special meaning in Judaism beyond the obvious as a means of reproduction. As such, I would disagree with some of the above sources claiming that the practice of placing one's hands under the thighs being a bowdlerization of hands on the testicles to be unlikely.

Rather, the placing of the hands over the penis according to some of the rabbinic literature is a reference to this being Abraham's most sacred feature (his penis being the living symbol of his covenant with God due to its circumcision). Or, it might just be literal that the master sits on the hands of the supplicant. Source.

Another in depth discussion of this concept can be found here.:

A popular claim also alleges that Greeks and Romans would touch their own testicles while swearing, however there is no evidence in support. The "testicle" theory argues that the testicles were used for oaths because they represented virility, power, and represented the man’s future generations, and the source of life. If so, perhaps that is the true "ancient Roman salute" (grasping one's own testicles), as the stiff-armed salute is definitely NOT an "ancient Roman salute."

The new theory asks whether the “inner thigh” posture (the "yarek oath" or "yarek prayer" or "yarek pledge") acknowledged the man’s circumcision. In Judaism the circumcision is the male’s covenant with God and is also called the “Covenant of Abraham,” because it began with the Patriarch Abraham. Two references to the “inner thigh” oath (above) refer to Abraham, whose circumcision would have been new and revered. Abraham would have circumcised his slaves, who later performed “inner thigh” oaths to Abraham.

It stands to reason that others who made similar covenant with God might consider their genitals to be sacred as well. And, they would be likely to consider it a portable symbol of their connection with God. (Particularly before the Gutenberg Bible brought the sacred word of God to the common man in a much less intimate form.)