Is there a moral problem with using useful medical information that was acquired by evil methods? For example, the Tuskegee Syphilis Experiments and the Luftwaffe Hypothermia Experiments were conducted without regard for the life or consent of the subjects, yet useful information was gained.
If such knowledge was found later by ethical means, would this make any moral difference to the prior use of this knowledge?
Let me begin by not mincing words: Such experiments are evil by Objectivist standards, involving absolutely heinous rights violations of innocent people. Anyone who would commission or call for such a thing is evil; anyone who would knowingly and willingly participate in committing such an atrocity is evil; the entire scene of such a crime is an evil manifestation of evil people permitted and even encouraged by an evil political system to do evil things. NONE of these should have happened or been permitted to happen, and ALL of the people knowingly and willingly involved deserve punishments up to and including the death penalty.
Bottom line? It is the depth of moral and criminal depravity to willingly encourage or undertake such an outrage in any capacity or degree.
Now, recall that the virtue of integrity is a recognition that the purpose of thinking is action. This principle stands as an explicit moral reminder to ourselves that if a flourishing human life is our goal, that we must never allow any breach between mind and body, between thought and action, between what we know we should do and what we actually do. (Common sources of this sort of breach would include social pressures we cave to in seeking to please others, our own nonsocial whims or emotions, and so on -- all of which would frustrate the pursuit of a human life.)
Taking the above together, the answer to the question is that absent any approval or sanction of its evil origin, there is absolutely no moral problem in the use of such knowledge. Just the opposite, in fact: were, say, a doctor to act as if he didn't know what he in fact knows about syphilis, whatever the source, then that would be immoral on the integrity front, and likely even criminal because it would involve a purposefully-negligent breach of his Hippocratic oath and contract with facility and patient.
However, we must never lose sight of the fact that if someone undertakes any such experiments -- no matter how useful they think the knowledge it could provide might end up being -- then that is not only completely immoral, but also criminal, and they should be treated accordingly by the rest of us. And anyone who would even murmur the slightest approval or encouragement of such an outrage is morally black, and should likewise be treated accordingly. (Notice how these are also an application of integrity.)
Thinking that the ends justify the means is a sure recipe for moral disaster.