The process of childbirth is considered a situation of illness, the recovery from which requires one to recite Birkat Ha'gomel. A woman who delivered a child must therefore come to the synagogue when ten men are present and recite the Beracha in the Ezrat Nashim (women's section). Although some have claimed that Birkat Ha'gomel may be recited in the presence of ten women, and does not require specifically a Minyan, Halacha follows the view that the recitation of this Beracha indeed requires the presence of ten men.
The Beit Yosef (commentary to the Tur by Rabbi Yosef Karo, author of the Shulchan Aruch) cites a view that a husband can recite Birkat Ha'gomel in his wife's stead after childbirth. This opinion, however, is not accepted as the final Halacha, and therefore the wife must personally come to the synagogue to recite Birkat Ha'gomel.
It has also been claimed by some that the process of pregnancy and childbirth, unlike other illnesses, is a natural condition that occurs very frequently, and hence does not require the recitation of Birkat Ha'gomel. This opinion, too, was rejected by the Halachic authorities. The Chesed La'alafim (work of Halacha by Rabbi Eliezer Papo, Bulgaria, 1785-1828) and Kaf Ha'chayim (by Rabbi Chayim Palachi, Turkey, 1788-1868), among others, explicitly rule that a woman must recite Birkat Ha'gomel after recovering from childbirth to express her gratitude to the Almighty for helping her recover. The Kaf Ha'chayim held that this Beracha should be recited no earlier than seven days after the birth, at which point the woman generally has recovered from labor. Chacham Ovadia Yosef, in his work Yechaveh Da'at, follows this ruling and requires that a woman recite Birkat Ha'gomel seven days after childbirth if she feels healthy.
Summary: A woman must recite Birkat Ha'gomel in the presence of ten men after the week following childbirth, so long as she has recovered from the childbirth.
