The life cycle of the Xenomorph as depicted in the Alien franchise bears almost no resemblance to the life cycle depicted in the movie Prometheus.
The xenomorph's life cycle is actually not explained in the first Alien movie, and only becomes clear in the sequel, Aliens. There, the cycle is presented as something slightly similar to an ant's, but not quite the same as any species that I know of.
A Queen xenomorph lays eggs. The eggs hatch into face-huggers. The face-huggers attach themselves to a host organism, usually a human, and implant a different kind of egg into the host's throat; the face-hugger then dies. This egg eventually hatches into a chest-burster, which grows into an adult xenomorph. I'm not sure if we ever learn whether any adult xenomorph can become a Queen, but it is at least plausible. Most of the time, however, the adult xenomorphs seem to serve as drones or slaves, protecting the queen and her eggs, collecting hosts, trapping them in the lair, or hive, and possibly serving other functions as well.
In the movie Prometheus, on the other hand, the life cycle is wildly different from this. David finds bottles of a strange, black liquid, which he surreptitiously places in Charlie's drink. Charlie consumes the liquid, has sex with Elizabeth, impregnates her, and becomes quite ill. The fetus, or zygote, or whatever you want to call it, is a monstrosity. She undergoes a procedure to remove the monstrosity, which apparently grows into a much larger creature very quickly. Near the end of the film, the monstrosity has become an enormous version of a face-hugger, and attacks one of the "Engineers". The Engineer's corpse later splits open, and something very similar to a xenomorph crawls out.
In both cases, one thing is constant: a xenomorph comes from an egg implanted into a host organism's throat by a face-hugger.
Nothing else is consistent between the two depictions, but it seems to be almost certain that a xenomorph won't pop out of your chest unless a face-hugger has already implanted an egg in your throat.
Note: This answer is based on the first two movies in the Alien franchise and the movie Prometheus. If there is some other information about the issue in other sources, I don't know enough about it to use it in my answer.