Is there any circumstance in which a private individual or
organization could undertake a criminal prosecution for a violation of
18 U.S. Code § 2441 - War crimes?
No.
Private prosecutions for federal crimes are not possible in the U.S.
The U.S. Constitution also requires a grand jury indictment prior to bringing any felony prosecution, and in this case, high U.S. government officials must approve bringing the charges as well.
In Linda R.S. v. Richard D., the Supreme Court held that "a private
citizen lacks a judicially cognizable interest in the prosecution or
nonprosecution of another". However, it seems impossible to violate §
2441 while acting in one's capacity as a private citizen, so it is not
clear that that principle would operate to bar a private prosecution
here.
You have totally misunderstood the quoted language. The quoted language has nothing to do with the capacity in which someone violated a criminal law, it has to do with the standing of someone to enforce a criminal law. Linda R.S. v. Richard D. holds that private citizens have no right to see that the criminal laws are enforced.
Linda R.S. v. Richard D. specifically holds that you can't bring an action for injunctive relief to force the government to enforce criminal laws. Castle Rock v. Gonzales, 545 U.S. 748 (2005), meanwhile, bars civil actions for money damages by people who are personally harmed by the failure of the government to enforce criminal laws or restraining orders.
If not, what sorts of injunctive relief or civil damages might be
available in U.S. courts, by virtue of someone having violated, but
not yet having been prosecuted for violating, this law?
If a war crime was committed against you, then you, or if you are deceased your estate (or next of kin in the case of some wrongful death statutes in lieu of your estate) can bring a claim for money damages in most cases (if there is not governmental immunity, which there often would be). This is because almost everything that is a war crime with a victim is also a tort which can be enforced in a civil action brought by the victim against the wrongdoer.
This could be done for victims of the crime who are not U.S. citizens where the war crime occurred abroad under the Alien Tort Statute:
The Alien Tort Statute ("ATS"; also known as the Alien Tort Claims
Act) is a common name for 28 U.S.C. § 1350. The ATS grants federal
district courts original jurisdiction over any civil action where an
alien sues for a tort “committed in violation of the law of nations or
of a treaty of the United States." Broadly speaking, it serves as a
statutory instrument for gaining universal jurisdiction over
violations of international law.
In many cases, where war crimes are committed by a government official in the U.S., there would be a cause of action for the victim of the war crime, either in a § 1983 action (for state and local officials), or in a Bivens action (for federal officials).
If you are not the victim of the war crime, you can't sue because you don't have standing to do so.
These theories of tort liability are entirely distinct from 18 U.S. § 2441.
