A verb to combine “to prove” and “to disprove”

Great question.

In fact, writers will often simply use both "prove" and "disprove" in this context. For example:

What does an experimental test of quantum contextuality prove or disprove? (link)

Experiments are controlled tests designed to prove or disprove a hypothesis. (link)

In fact, these two simple and randomly chosen examples both contain a single word that can refer to the concept of proving and disproving: test.

Here's the relevant verb definition from Merriam-Webster online:

to put to test or proof

The verb definition actually uses the noun definition, so let's have a look at that as well (or at the relevant definition), because it contains more precisely what you're looking for:

a critical examination, observation, or evaluation : trial; specifically : the procedure of submitting a statement to such conditions or operations as will lead to its proof or disproof or to its acceptance or rejection

If you want more generally to state that something is not relevant to an argument, then you can use any one of the words synonymous with irrelevant (which is the relevant word that you, in fact, used in your post):

not germane

not related

not applicable

Or, my favorite, because of its dual meaning, both of which may be pertinent if you're peeved that someone is offering an opinion or statement that is neither relevant nor polite:

impertinent

The adjective inconclusive comes immediately to mind.

ADJECTIVE

Not leading to a firm conclusion or result; not ending doubt or dispute:

Inconclusive is different than irrelevant (and many of its synonyms). While something irrelevant is disqualified from impacting a conclusion, something inconclusive may impact the conclusion, but is insufficient to form the conclusion in itself.

"The circumstantial evidence was inconclusive in the murder investigation, but the price of eggs in China was irrelevant."

It seems the positive action of proving is not "verbally" comparable to the inconclusive status of the argument. The verb conclude "combines" prove and disprove by defining an action at the intersection of prove and disprove. In any frame of reference, proving p as true simultaneously disproves p as false. One could coin antonyms of conclude, like contraclude to mean "not closing (the investigation)", but we are simply stating that the argument has not yet "combined" prove and disprove in a conclusion.

This is a very interesting convergence of philosophy and language. You can imagine yourself in a plane looking down on two Interstate highways. From the perspective of your question, they seem to "intersect", but from every other perspective, they clearly pass over one another without touching.

Since p and the truthfulness of p are separate entities,

The simplest way to express the concept you are considering is:

"That leaves p (or the truthfulness of p) open for investigation (or discussion)."

A single word that provides the sense you want could be one of resolve, determine, decide, or settle. They are all approximate synonyms in this context.

The definition of resolve is:

: to find an answer or solution to (something) : to settle or solve (something)
: to make a definite and serious decision to do something
: to make a formal decision about something usually by a vote
_{(Source: Merriam-Webster)}

So, in your sentence:

That line of reasoning, while interesting, does nothing to resolve the proposition p, and is therefore tangential to our discussion.

In the legal arena, one would say that the (discussion, fact, statement, etc) is not dispositive, in that you are not able to "dispose" of the issue, i.e. resolve it one way or the other. One is often hoping for dispositive evidence in a trial, since it will allow one to close the case.

http://legal-dictionary.thefreedictionary.com/Dispositive+Fact

http://www.law.cornell.edu/wex/dispositive_fact