Let me respond to your question with a question: how would you, in your mind, go about testing or falsifying a historical argument?

In general historians do think of themselves as ultimately being empirical in methodology (we base our claims on facts) but there is no strict criteria for falsifiability that is imposed generally speaking. (If you wanted to speak in the language of philosophy of science, my experience is that most historians are verificationist in their approach to theses — they look for evidence for them and don't worry if there is no possibility of evidence against them.)

Is this an issue? Well, I don't know. It depends on what kinds of historical claims we are talking about. If one is talking about strictly factual claims, then ideally, yes, it would be nice if they were falsifiable. However there are severe limitations on evidence bases so there is going to be a lot that is just unlikely to be falsifiable even in principle.

But a lot of historical claims are not narrowly factual, and are based on broad interpretations and syntheses. Most of these are unlikely to be strictly falsifiable. History, unlike science, does not always seek to reduce a natural phenomena to the simplest of elements. It is also strongly constrained by the fact that it cannot run "experiments" — it is a strictly historical form of empiricism.

I think in general you would find most historians would not proposed falsifiability as a minimum methodological requirement for historical claims. They do generally require some kind of evidence, but it can be of a non-falsifiable character (again, like verificationism). But to impose falsifiability would probably be too deep of a cut for the types of historical claims that could be made. This is not to say that all historical claims are not falsifiable — some quite clearly are.

At the base of the question is some concern about whether the factual status of historical claims equals the factual status of claims in the natural sciences. I think very few historians would want to make that claim. History is not a natural science. It requires much more interpretation, much more guessing, much more fuzziness. This is because the complexity involved in macro- and microscopic interactions between human endeavors over time (which might be one excessively scientistic definition of "history") is excessively large, and it has traditionally resisted reductionist arguments that are more common (and have been very successful) in the sciences. This is not to say that one should assume history has a special methodological status, just that on some kind of sliding scale between "hard" and "soft" and "social" sciences, history is on the "hard science approaches don't work well here most of the time" end of the scale.

EDIT: To clarify terminology among people not well-versed in the philosophy of science:

• Verificationism is the belief that a theory is a good theory if you can find evidence proving it correct.

• Falsification is the belief that a theory is a good theory only if the possibility exists of proving it incorrect. (You don't have to actually try to prove it incorrect, it just has to be imaginable that evidence could prove it wrong.)

Verificationism was pushed by the initial Vienna Circle positivists as a criteria for science. Karl Popper said this was dumb because it is easy to find positive evidence for empty theories (e.g. I believe that the world is governed by an invisible unicorn; I believe that the invisible unicorn would like things to work out well for me; whenever something works out well for me, I see it as evidence of the existence of the unicorn) and instead said falsification was a better criteria (e.g. If there is no possible evidence that would refute the existence of my invisible, benevolent unicorn, then it is not a good theory).

Many scientists like Popper's approach to epistemology because it allows you to throw out some obviously bad theories (e.g. invisible unicorns, or Creationism, or Intelligent Design, etc.) as "not science" (pseudoscience). However many philosophers and historians of science have pointed out that falsification is not quite the solution that it seems to be, and there are a number of theories we generally assume are "scientific" that do not easily pass it (e.g. string theory, even Darwinism is a tricky case). And the act of actually proving a scientific theory wrong is almost always more complicated than a simple experimental observation.

I say historians seem to be verificationists because they will entertain any theory there see evidence for. They do not require one to imagine that there is a way to disprove it prior to proposing it. There are indeed some historical theories and theses that are so vague that it is hard to imagine evidence being decisive. On the other hand, most historians do tend to think of their theories as empirically based, which means they think that evidence could be found to disprove them, so in some sense they are very weak falsificationists. Overall historians worry less about these kinds of things than scientists, I might suggest, because pseudo-history is a lot less common and less persuasive than pseudo-science is.