The Church is strongly in favor of stem cell research. Indeed, it has funded and even led conferences in attempts to find medical solutions via stem cells. The Church has almost always supported science, and even the Galileo affair was really an argument about whether he had actually proven his claims about relationship of the earth and the sun scientifically. (Many secular scientists believe he had not.)
The Church objects to killing fellow members of the species homo sapiens in order to get the benefit of embryonic stem cell research science. Nazi Germany showed us that science and medicine need to be disciplined by morality, and the Catholic principle that “it is always wrong to aim at the death of an innocent person” is a strong bulwark against the powerful trampling on minority and vulnerable populations. Happily, the best stem cells (like the “pluripotent” cells found in amniotic fluid) do not come from embryos, so we do not need to kill some humans in order to heal others with stem cells.
From Charles C. Camosy and the Busted Halo Question Box.