I don't think stem cell manufacture would be considered a part of the chemical industry.
Currently, the main method of obtaining pleuripotent embryonic human stem cells is from the inner cell mass of ~8 day old embryos. At this point the embryos are in a stage of development called the blastula or blastocyst, which consists of the an outer layer of cells, the trophoblast (which will eventually become the placenta and amniotic sac), surrounding a mass of cells which will eventually develop into a fetus. This inner cell mass is what researchers isolate in order to obtain stem cells.
Since there are no synthetic means of creating the embryos, sperm, or egg neaded to produce a blastula, this would not fall under chemical industry. It would probably be best classified as biotechnology.
Other types of stem cells can be obtained from other sources (such as adult stem cell tissues like bone marrow), umbilical cord cells, or aborted fetuses, but none of these sources would fall under chemical industry either. Currently, a cell is much too complicated to produce synthetically.