Short answer
Yes. In Talmudic law, a betrothal through intercourse with a girl aged three years and one day, with her father's consent, was legally valid. It made her legally a married woman. The Mishnah, the first written collection of the rabbis' laws (about 200 CE), sets that age (Niddah 5:4). The Talmud repeats the rule in Sanhedrin 55b, and discusses it in Niddah 44b-45a. Below that age, Ketubot 11b says, an adult's act with a girl is "nothing" in law, so she keeps the status of a virgin. The Talmud also says a father may not betroth a young daughter "until she grows up and says: I want so-and-so" (Kiddushin 41a). The Rema was a 16th-century rabbi. His note on the standard code of Jewish law says that in exile the custom became to betroth young daughters, "and so is the practice". Islamic law forbids married life before a wife is physically ready. And Muslims keep the minimum age of marriage set by the law of the country they live in (17:34). The site answers the charge about Aisha on its own page. Islam blames no one living today for an old text (6:164).
Full explanation
What the Mishnah says (Niddah 5:4)
The Talmud has two layers. The Mishnah is the first written collection of the rabbis' laws, compiled about 200 CE. The Gemara, the second layer, records later rabbis' debates on the Mishnah. It was finished about 500-600 CE. A reference such as Ketubot 11b names a part of the Talmud and a page: page 11, second side. Mishnah Niddah 5:4 sets three years and one day as the age from which intercourse with a girl has legal effect. From that age she can be betrothed through intercourse, with her father's consent, as Maimonides and the commentators explain. In Jewish law this betrothal (kiddushin) is already a binding marriage: from then on she is legally a married woman, and it takes a divorce to end it. A man who has relations with her when she is married is guilty of adultery. Any man the Torah forbids to her as a close relative is put to death for it. She is not punished, because she is a child. Below that age the act has no such legal effect.
The Mishnah (compiled c. 200 CE) sets three years and one day as the age from which intercourse with a girl has legal effect. From that age she can be betrothed by intercourse, with her father's consent, as the commentators and Maimonides explain. A man who has relations with her when she is married is liable for adultery. Any man the Torah forbids to her who has relations with her is put to death, and she is exempt. Below that age the act has no such legal effect.
The Talmud's discussion (Niddah 44b-45a)
The Talmud's discussion of this rule (Niddah 44b-45a) debates exactly when it starts. Then it tells a story. A Roman noblewoman asks Rabbi Judah the Prince, who compiled the Mishnah, at what age a woman is married. He answers three years and one day, and twelve years and one day for pregnancy. She says she married at six and bore a child at seven. The Talmud doubts the pregnancy. It cites a ruling on a young wife's safety. Rabbi Meir, a 2nd-century rabbi, rules that a wife aged eleven to twelve uses a contraceptive, lest a pregnancy kill her. The Sages leave it to Heaven's mercy.
The Gemara on Mishnah Niddah 5:4. Baraitot record Rabbi Meir and the Sages differing over the exact starting point of the Mishnah's rule on legal effect. The Gemara then asks, for questions of ritual purity, how the body of a girl under that age heals. A baraita tells a story (aggadah): Yusteni, a Roman noblewoman, asks Rabbi Judah the Prince at what age a woman is married. He answers three years and one day, and twelve years and one day for pregnancy. She says she married at six and bore a child at seven. The Gemara doubts the pregnancy, citing a baraita on a minor wife's safety: Rabbi Meir rules that a wife aged eleven to twelve uses a contraceptive lest a pregnancy kill her, and the Sages leave it to Heaven's mercy. A second story: Rabbi Akiva's students hold, as a law given to Moses at Sinai, that a woman to whom this happened before the age of three may still marry a priest.
Ketubot 11b and Sanhedrin 54b-55b
Ketubot is the part of the Talmud about the marriage settlement, the sum a husband owes his wife under the marriage contract. A virgin's settlement was two hundred zuz (silver coins), and other women's one hundred. The Mishnah (Ketubot 1:3) lists women who keep the higher sum. In Rabbi Meir's ruling, one is a woman with whom an adult had relations when she was a child. Rava, a leading Babylonian rabbi of the 4th century, explains this on page 11b. An adult's act with a girl under three is "nothing" in law. Below that age, he says, the body heals, as an eye that is touched sheds a tear and recovers. So she still counts as a virgin. Sanhedrin is the part of the Talmud on courts and crimes. Page 54b asks from what age of a boy the Torah's death penalty for relations between males applies. Rav, a leading rabbi who died in 247, says nine years. Shmuel, another leading rabbi, who died about 254, says three. On page 55b Rav Yosef, a 4th-century rabbi, quotes Mishnah Niddah 5:4: from three years and one day a girl can be betrothed through intercourse.
Halakha on the ketubah, the marriage settlement a husband owes his wife: two hundred zuz for a virgin, one hundred for others. Rabbi Meir rules that three women keep the settlement of two hundred: one with whom an adult had relations when she was a minor; one who, as an adult, had relations with a minor boy; and one injured by a piece of wood. The Sages give the injured woman one hundred.
On Mishnah Ketubot 1:3 and the amount of the marriage settlement. Rav holds that an act by a boy under nine leaves a woman like one injured by wood; Shmuel disagrees. Rava explains the Mishnah: an adult's act with a girl under three is "nothing" in law, because below that age the body heals, as an eye that is touched sheds a tear and recovers. So she keeps the status of a virgin and the full settlement. The rest of the folio discusses when a settlement is reduced or lost.
On the Torah's death penalty for relations between males (Leviticus 18:22, 20:13) and with animals. A baraita says that for relations with a male the Torah did not treat a young boy like an older one. Rav explains: an act with a boy under nine is not counted in law like an act with a boy of nine; Shmuel sets the line at three. A baraita follows Rav: a man who has relations with a male of nine years and one day is liable.
In a debate on another point of the same criminal law (whether an animal involved in a sexual crime is put to death), Rav Yosef quotes Mishnah Niddah 5:4. From three years and one day a girl can be betrothed by intercourse, and a man who has relations with her when she is married is liable for adultery. Any man the Torah forbids to her who has relations with her is executed, and she is exempt. The Gemara explains that the Torah spares her out of pity, because she is a minor. Rava quotes the parallel rule for a boy of nine (Mishnah Niddah 5:5). The question is left unresolved.
What the Talmud tells fathers (Kiddushin 41a)
Kiddushin, betrothal, is the first stage of a Jewish marriage. The Mishnah lets a father betroth his daughter "when she is a na'arah", a girl of twelve who has begun puberty. From this the Talmud draws the lesson: not while she is still a child. Rav rules that it is forbidden "until she grows up and says: I want so-and-so". Maimonides, who died in 1204, writes that a father keeps the legal right, but it is not fitting to use it. The Shulchan Aruch (1565), the standard book of Jewish law, calls it a mitzvah, a commandment, not to do it. The Rema, a 16th-century rabbi, added a note to it. The note says that in exile the custom became to betroth young daughters, for fear that a dowry or a good match would be lost, "and so is the practice".
The Mishnah says a man may betroth his daughter "when she is a na'arah", a girl of twelve who has begun puberty. The Gemara infers: when she is a na'arah, yes; when she is a minor, no. This supports Rav, as Rav Yehuda reports him (some say it was Rabbi Elazar): it is forbidden for a man to betroth his daughter while she is a minor, until she grows up and says, "I want so-and-so". On the same page Rav also forbids a man to betroth a woman before he has seen her, lest she become hateful to him, since the Torah says "love thy neighbour as thyself".
The Mishnah (compiled c. 200 CE) lets a man give his daughter in betrothal while she is a na'arah, a girl of twelve.
Though a father has the legal right to betroth his daughter to whom he wishes while she is a minor or a na'arah, "it is not fitting to do so". The Sages commanded that a man should not betroth his daughter while she is a minor, until she grows up and says, "I want so-and-so". Nor is it fitting for a man to betroth a minor girl. Nor should a man betroth a woman until he has seen her and she pleases him.
Main text: it is a mitzvah (a commandment) not to betroth one's daughter while she is a minor, until she grows up and says, "I want so-and-so". The Rema's gloss: some say that in this time the custom is to betroth our minor daughters, because we are in exile and do not always have enough for a dowry, and we are few and do not always find a fitting match (citing the Tosafot); and so is the practice.
Rav forbids a father to betroth a minor daughter because, had she been grown, she might not have agreed. The Tosafot add that "now" the custom is to betroth daughters even as minors, because the exile grows heavier every day: a father who can give a dowry now may not be able to later, and his daughter would be left unmarried for ever.
Islamic law, and the charge about Aisha
Islamic law forbids married life before a wife is physically ready, and, as a general rule, a woman must be asked. Today Muslims also keep the minimum age of marriage set by the law of the country they live in, wherever that is (17:34). The site answers the charge about Aisha on its own pages. "How old was Aisha when she married the Prophet (ﷺ)?" gives her own account and sets out that law. "Does Qur'an 65:4 allow marriage to girls who have not reached puberty?" shows what that verse does and does not say. Islam blames no one living today for an old text: "no bearer of burdens can bear the burden of another" (6:164).
Al-Nawawi gives the ruling of Malik, al-Shafi'i and Abu Hanifa that a wife is handed to her husband only when she is able to bear intercourse, which differs from girl to girl and is not fixed by an age, and he calls this the correct view.
Aisha asked the Prophet (ﷺ) whether women should be asked for their consent to marriage. He said yes; and when she said a virgin may be too shy to speak, he said her silence is her consent.
Khansa bint Khidam, who had been married before, reports that her father married her off against her wishes, so she went to the Prophet (ﷺ), and he declared the marriage invalid.
Aisha reports that the Prophet (ﷺ) married her when she was six, that the marriage was consummated when she was nine, and that she then lived with him for nine years.
“Say: "Shall I seek for (my) Cherisher other than Allah, when He is the Cherisher of all things (that exist)? Every soul draws the meed of its acts on none but itself: no bearer of burdens can bear the burden of another. Your goal in the end is towards Allah: He will tell you the truth of the things wherein ye disputed."”
“Come not nigh to the orphan's property except to improve it, until he attains the age of full strength; and fulfil (every) engagement, for (every) engagement will be enquired into (on the Day of Reckoning).”
Perspectives
Traditional Jewish reading
These passages are Jewish law about legal status. They say from what age an act has consequences for marriage, for ritual purity and for criminal punishment. The Oral Torah, the teaching handed down by word of mouth, was given with the written Torah at Sinai. The rabbis of the Talmud forbade a father to betroth a young daughter.
Jewish law forbids a father to betroth a young daughter until she grows up and chooses. Islamic law did not forbid it.
Rebuttal
Kiddushin 41a forbids it. Yet in Talmudic law such a betrothal still counted. And the Rema records that betrothing young daughters became the custom in exile, "and so is the practice". Islamic law forbids married life before a wife is physically ready. The page on Aisha's age gives her own account, and that law in full.
These rules are part of the Oral Torah, which Orthodox Jews hold was given with the written Torah at Sinai. And they protect the girl.
Rebuttal
The same rules make a betrothal through intercourse valid from three years and one day. Islam holds that Israel's later tradition mixes true teaching with what men added. And the Qur'an is the criterion over it (5:48).
The Islamic perspective
In Talmudic law a girl of three years and one day could be betrothed through intercourse. The Talmud also forbade a father to betroth a young daughter. Islamic law forbids married life before a wife is physically ready. The charge about Aisha is answered on the site's own pages.
In Talmudic law, from three years and one day, a girl could be betrothed through intercourse, with her father's consent.
The Talmud forbids a father to betroth his daughter while she is a minor. He must wait until she grows up and chooses.
Islamic law forbids married life before a wife is physically ready. And as a general rule it requires that a woman be asked.
Israel's later tradition mixes true teaching with what men added, and the Qur'an is the criterion over it (5:48).