The metaphor is wider than Ishmael allows in the book. Birth, at the time the novel was written, was a dangerous business. The passage in question makes clear that this birth is equally dangerous to the participants. Both Tashtego and Queequeg almost drown and the valuable head is lost in the depths of the sea.
The crew of the ship is all-male yet the metaphor of birth here is clearly feminine and it seems to fascinate the whalemen. This is not the only time that birth is used as a metaphor in Moby-Dick. A more horrific "masculine" birth is used several times to describe how the male imagination has "birthed" Moby-Dick itself.
half-formed foetal suggestions of supernatural agencies, which eventually invested Moby Dick with new terrors unborrowed from anything that visibly appears
Ahab's desire to slay the whale is described in similar terms:
That purpose, by its own sheer inveteracy of will, forced itself against gods and devils into a kind of self-assumed, independent being of its own. Nay, could grimly live and burn, while the common vitality to which it was conjoined, fled horror-stricken from the unbidden and unfathered birth
These two conceptions of birth foreshadow the ending of the novel. First, the Pequod is sucked down into a womb-like vortex, entombed in the sea. This is the disastrous consequence of Ahab's warped, masculine "birthing" of his obsession.
And now, concentric circles seized the lone boat itself, and all its crew, and each floating oar, and every lance-pole, and spinning, animate and inanimate, all round and round in one vortex, carried the smallest chip of the Pequod out of sight.
Then the feminine ship Rachel arrives, like a new mother, to rescue the survivors, including Ishmael.
It was the devious-cruising Rachel, that in her retracing search after her missing children, only found another orphan.
It's also noteworthy that at all three harpooners on the ship are non-Christian "savages". The rescue of Tashtego is not so much a birth as a "rebirth", and as such mirrors a resurrection from the dead, a miracle performed only by Christ. This passage shows that Christians are not the only progenitors of such miracles. Instead, concern for our fellow man allows any of us to perform the miracle of rebirth in rescuing some poor unfortunate in distress.
Parts of this answer reference the paper 'Hunting and Writing the Whale: Masculine Responses to the Maternal in Herman Melville' by Seth A. Hagen