Safe Responsible Ethical Scientific Endeavors Assuring Research for Compassionate Healthcare Act or the Safe RESEARCH Act This bill prohibits the National Institutes of Health (NIH) from conducting or supporting research on human fetal tissue obtained from an abortion procedure. It also imposes other restrictions on human fetal tissue research. Under the bill, the NIH may conduct or support research on human fetal tissue only if it was obtained from a stillbirth. Current law allows research on tissue from stillbirths or from spontaneous or induced abortions. The bill also applies informed consent and other requirements applicable to research on the transplantation of human fetal tissue for therapeutic purposes to all research with human fetal tissue. This includes research on stem cells and other human fetal tissue alternatives. Additionally, human fetal tissue used for this research must be obtained in accordance with state anatomical gift laws. These laws govern organ and tissue donation for therapeutic, research, and other purposes. Some states already apply anatomical gift laws to human fetal tissue donation. Currently, executive branch officials may not prohibit the NIH from conducting or supporting research on the transplantation of human fetal tissue for therapeutic purposes. The bill repeals this limitation on executive branch officials. It also repeals a requirement that the NIH fund certain proposals for this research.