No. It's not likely. Did Bugfartboy die after giving birth to a Pum-Kin, which came out of his womb looking very blue, because Pum-kins' respiratory system is different from that of a Bugfatboy, and could not gain much oxygen from Buggy. Congratulations, dead Bug! You're a mother!
Actually, it has been seen since, and there are large macromolecules called "proteins" that group together into working organelles every once in a rare while (about once in 10 days), which in turn are surrounded by a bunch of fats and proteins grouped together in a naturally forming item called a "lipid bilayer" rarely (about every 100 years or so). Most of the time, these new cells that show up are totally outcompeted by the life we have around here today.
Which is why we don't seem to see evolution in progress today; a fish won't turn into a frog, because in it's half-frog stage of evolution, it's going to be a sucky fish, and an even suckier frog, and the frogs that have had millions of years to perfect being frogs kick their arse.
After the first cell, any further life generated from random interactions was redundant, and the species which had had a chance to adapt remained dominant.
None of this happens overnight, but it happens a lot faster than you would think. It only takes a few hunred million years to get self-replicating organic molecules, and only one more billion years to get a cell. Then, a couple of billion years to get multi-cellular life to occur, gradually working its way up from symbiotic relationships, to dependency, to mutual reproduction. After that, it just takes off like an increadibly slow rocket!
