This is an interesting theory, but as others have mentioned, I doubt that it always works if a mother has multiple children, all of whom were born naturally.
Unfortunately, the number of children born "naturally" has decreased sharply throughout the years, what with the apparent rise of infertility and the accompanying IVF and other fertility treatments; even if such an "artificially"-conceived child were not to be born via Cesarean section or induction, would you consider this child to have had a "natural" birth?
There are quite a few other scenarios that I can think of which test the boundaries of this definition of a "natural birth," such as if the mother needed the assistance of pitocin if labor had stopped--this happened to my mother when I was born--or if she had requested an epidural or other pain-relieving medication but her labor had not been induced, and she proceeded to deliver "naturally" after the medication had been administered.
The strictest definition would probably apply to an incredibly small percentage of the infant population, but it would be interesting to study this small percentage to see if there were any significant correlation between these babies' rising signs and the mother's moon signs.
My moon is in Aries and I so see my mom as a little powerhouse. She kind of scares me she has so much energy!
Although my brother and I have our natal Moon in different signs, we share similar Moon influences; I have an Aries Moon and he has an exalted Mars conjunct a debilitated Capricorn Moon. One would assume that our mother was a fire sign, or else had some other strong Mars influence, but she's a Libra with a Libra stellium, which includes a debilitated Mars in Libra! Unfortunately, I don't have her exact time of birth because it was not recorded on her birth certificate, but I know that I certainly perceive her as a "little powerhouse." That's a very apt description, since she is under five feet tall; don't mess with her, though!
Arian Maverick