Why is there no Bris Milah for girls?

Question

The title says it all, for the most part, but given that the Bris Milah is a way of bringing a newborn boy into the covenant with God, why is there no parallel/similar ritual, or commencement, as it were, for girls?

Answer 1

The answer to that question is more difficult than appears at first glance. One might have be said "gloves for snakes? combs for hairless?"
One of the assumptions made by the question is that Brit milah is a supplement. If it is lacking, this is an under privilege.
But following the Rambam (the Guide for the Perplexes III, 49), it is as a quasi-amputation.

As regards circumcision, I think that one of its objects is to limit sexual intercourse, and to weaken the organ of generation as far as possible, and thus cause man to be moderate. Some people believe that circumcision is to remove a defect in man's formation; but every one can easily reply: How can products of nature be deficient so as to require external completion, especially as the use of the fore-skin to that organ is evident. This commandment has not been enjoined as a complement to a deficient physical creation, but as a means for perfecting man's moral shortcomings. The bodily injury caused to that organ is exactly that which is desired; it does not interrupt any vital function, nor does it destroy the power of generation. Circumcision simply counteracts excessive lust; for there is no doubt that circumcision weakens the power of sexual excitement, and sometimes lessens the natural enjoyment: the organ necessarily becomes weak when it loses blood and is deprived of its covering from the beginning. Our Sages (Beresh. Rabba, c. 80) say distinctly: It is hard for a woman, with whom an uncircumcised had sexual intercourse, to separate from him. This is, as I believe, the best reason for the commandment concerning circumcision. And who was the first to perform this commandment? Abraham, our father! of whom it is well known how he feared sin; it is described by our Sages in reference to the words, "Behold, now I know that thou art a fair woman to look upon" (Gen. xii. 11).

To reach spiritual perfection, we must abandon physic perfection. It is linked to foreskin, to viril problem only, women do not need this. Milah is a treatment against viril physic passion, to protect spiritual devlopment.

Answer 2

There actually is a naming ceremony for girls, called zeved habat.

Zeved is a gift (cf. Lea's naming of Zevulun in Bereshit 30:20). It is a way to express' parents hakarat hatov (gratitude) to God for the gift of a girl. This is quite common in Sephardic communities in France (personal experience) and surely elsewhere as well. There is a traditional prayer in the mainstream sephardic French siddur.

See for instance here regarding Syrian communities.

The zeved habat goes back many generations, and is still celebrated today. In the Syrian Jewish community of Brooklyn, New York, on the first Shabbat after a girl is born, her father, along with his father and father-in-law, are called to the Torah. The baby and her mother are usually in attendance, but since this is a primarily Orthodox community, only the men are called up for aliyot, the honor of saying the blessings before and after the public reading of a section of the Torah. The father says the blessings over the Torah reading twice–once on his own merit and once in honor of his daughter –and the grandfathers each have an aliyah as well.

The rabbi offers the family congratulations on their new arrival and offers a misheberach, a prayer for the girl’s well-being. Then the words “avi habat,” or “father of the daughter,” are called out. That is the congregation’s cue to start singing traditional songs for welcoming girls. The songs, based on poems dating back to 14th and 15th century Spain, are known as pizmonim. Women and men join in together.

See also here

Though several sources claim that Simchat Habat it is a “new” or revived custom peculiar to the United States, similar ceremonies have taken place in Sephardic and Ashkenazic Jewish communities since the Middle Ages. The common features of these different ceremonies are an “aliya” (calling to the torah) of the father in the synagogue, with announcement of the name of the child, and a celebration for family and friends at home.

[...]

A rabbi or Hacham presides over the Zeved Habat ceremony and several prayers and verses are recited. The mother recites Birkat Hagomel, the prayer of deliverance in honor of the safe birth.Song of Songs 2:14 is recited. If it is a first daughter, Son of Songs 6:9 is recited as well. Other prayers and recitations may include (Psalm 128 and the Priestly Blessing, Birkat Kohanim). The main recitation is the Mi shebberach prayer for naming the baby girl.

Answer 3

There was a Brith Milah for girls amongst one Jewish group, though there is no way to know whether or not this practice was of Jewish origin, or was of some African pagan origin that infiltrated the Jewish community.

The Ethiopian Beta Israel community had a tradition of female circumcision. i don't know if there was a celebration marking the occassion, or if it was done as just common practice (like a doctor performing a male circumcision in the hospital). Thankfully upon migrating to Israel, this practice has almost (if not completely) died out. The articles in question make it appear that the practice was about limiting women's sexuality rather than a covenant with God, but the articles listed aren't exactly trying to find a religious reason for what is considered a very barbaric practice.