No, reincarnation is not open to empirical validation:
The problem of how life after death can be scientifically studied is not a simple one. Parapsychologists believe progress can be made in a scientific fashion, but they have received doubt both from the skeptic community and from sympathetic, neutral philosophers of science. It is this latter criticism I will focus on. The remainder of this answer is an extremely brief and hopefully accurate discussion of Michael Sudduth's new book A Philosophical Critique of Empirical Arguments for Postmortem Survival (Palgrave Macmillan, 2016).
The example of Ian Stevenson's work on reincarnation is pretty straightforward: he proposes to simply find case studies of alleged reincarnations who possess factual information about their past lives that they couldn't have come across by normal means. This would seem to be evidence of some kind of paranormal activity, and indeed, much of the criticism of Stevenson has focused on trying to disprove his dozens of examples.
However, what Stevenson provides is data, not a hypothesis. The "survival hypothesis," the statement "The soul survives death", does not cause us to expect any this-worldly phenomena in itself. We could be living in an orthodox Christian world where the future states of the soul do not affect us in any way. That doesn't make the claim unusual; according to the Durhem-Quine thesis, there are very few scientific claims that can be falsified by themselves. To restate Wikipedia's example, Galileo seeing rings around Jupiter could be explained by Jupiter being surrounded by a dozen small moons and thousands of tiny objects in a ring, or it could be explained by a scratch on the lens of Galileo's telescope, as one of the other intellectuals of the day suggested. So, a truer statement of Galileo's hypothesis is actually "Jupiter has rings and I can see them perfectly well through my telescope". The second half of this is called an auxiliary hypothesis. It's not essential to the main hypothesis but we need it to get from the data we currently have to the main hypothesis.
Similarly, the survival hypothesis in a more testable form must make more specific claims that cause us to expect data, like "The soul survives death and goes to Heaven, and people can experience this themselves through near death experiences," or perhaps "The soul survives death, reincarnates sometimes on Earth, and retains at least some of its memories, and we can confirm this and rule out cold reading through carefully controlled interviews." The latter is closer to Stevenson's actual view.
But we are not done with auxiliary hypotheses yet, because at some point they lose explanatory power. For example, if Stevenson were to narrow down his claim to "I myself spoke to a reincarnated person once, in a unique event that cannot be duplicated," that's not particularly helpful for our lives. In the more general form he gives in his books, Stevenson's claims can be explained in all sorts of other ways:
- Children like making up stories and sometimes hit on several real facts about other people through totally random chance.
- Children can access the memories of relatives of a deceased person through some unknown mechanism. (Living psi)
- Memories drift around in the ether at death and sometimes fall into other people's heads totally independent of the question of survival of the soul. (René Guénon's "psychic residues")
- Stevenson was systematically deceived by his interpreters, or for some other reason didn't see what he thought he saw. (Cartesian demon)
- Regardless of what country you're in, for some reason -- pursuit of fame, confirmation of community beliefs, etc. -- people have a strong inclination to fake reincarnation-like incidents.
None of these alternative explanations actually falsify the claim "The soul survives death, reincarnates sometimes on Earth, and retains at least some of its memories, and we can confirm this and rule out cold reading through carefully controlled interviews." They just tell us we haven't truly tested it yet.
But alas, that tells us that we need additional auxiliary hypotheses which reduce the explanatory power of the survival hypothesis so much that there's no particular reason to accept it as an explanation of Stevenson's data over any of the other hypotheses. In particular the living psi hypothesis would seem to cover any anomalous data that could possibly be claimed to support the survival hypothesis. So according to Michael Sudduth's book, the argument that "the soul survives death" necessarily has no explanatory power, regardless of the more specific data it is meant to analyze, or the specific test meant to prove or disprove it. There is no way to state it in a way that becomes scientifically valuable.
Sudduth concludes that the survival hypothesis is not a scientific hypothesis, meaning that it can't be evaluated empirically. This is the problem with much of parapsychology: experiments may suggest to a reader that something weird is going on, but it's hard to turn "something weird" into a scientific hypothesis, especially when it's an anomaly that is not systematically reproducible and varies its nature widely between experiments. Such deficiencies reduce the explanatory power of any hypothesis significantly.