Funny - I went into labor (water broke) a month early, on a new moon. Nothing there.
Since the moon affects the tides daily (two high tides and two low tides) based on the moons daily orbit around the earth, I find it hard to believe that there is a monthly effect on bodies of water. Also, the moon has exactly the same amount of pull whether it's full or new because its mass doesn't change, and gravity is a function of mass. All that is different about a full moon vs. a new moon is the amount of sunlight we see reflecting off the moon. Also, tides occur because of the pure volume of water in the ocean. While there is a pull on smaller bodies of water, like lakes and ponds, the effect is small due to the volume of water being pulled. The total amount of water in the uterus is not that great - which again supports the hypothesis that the moon would not have an effect.
There is something that is called a neap tide, when the tides are higher because the moon and the sun are "lined up" and so pull in the same direction. In that instance, the tides are higher (or lower) because two bodies are pulling in the same direction and thus have more overall pull. However, I'm not aware that these correspond with a full moon necessarily.
In science, there is an adage that correlation does not equal causation. Just because something is correlated, like births and the full moon, doesn't mean that a full moon causes the increase in births.
If it is true that births increase on a full moon, I think that there is something else there - not the pull of the moon. Perhaps it's simply that 38 weeks prior to that is also a full moon and people tend to get romantic on a full moon? Or perhaps it's that it's a new moon 38 weeks before that and people are bored on a new moon?