Yes, the Catholic Church does indeed teach against stem cell research, but only that research that would damage or destroy human life (
embryonic stem cell research). It does advocate
adult stem cell research--adult stem cells can be harvested without destroying the human body (they could use fat cells even), and adult stem cells I believe have produced 9 cures.
Embryonic stem cell research, the research that destroys human life, has produced absolutely 0 cures and also has tended to
cause problems instead of
fix them. Even if they caused cures, it would be no excuse to destroy human life, because human life is sacred.
