From a balance point of view, you will have paid the cost (of not having a shield or second weapon equipped in your off hand). So that's OK.

In terms of rationale, watch someone throwing a javelin. The off hand is used for balance and additional momentum. There is definitely more force when the off hand is free*. So that's OK too.

I'd allow it.

Note that is not a RAW interpretation, due to the wording of the Versatile property specifically mentioning improved damage to a melee attack only. I may still allow it, so will leave the answer here. As far as I can see the melee-attack only constraint is in v0.3 of the online rules, but I cannot search further back. If anyone has older versions and can verify whether/when the wording has changed, I am happy to update the caveat just for accuracy's sake.

* yes this is a little facile. Don't look too hard at the narrative argument. Don't sweat the rationale to the point that a degree in Biomechanics is required. Don't ask me about hit points :-)

The answer to your question is No, using versatile weapons with two hands does not disqualify them as monk weapons.

But not because of any reason that made you ask the question, because your whole question is wrong as it's clear from your question you haven't understood the paragraph in question.

The paragraph in question on pg. 78, PHB says (emphasis mine):

At 1st level, your practice of martial arts gives you mastery of combat styles that use unarmed strikes and monk weapons, which are shortswords and any simple melee weapons that don't have the two-handed or heavy property.

Let's look at what that paragraph really says.

The first thing it does is stating that at 1st level a monk has a master of different combat styles using unarmed strikes and monk weapons.

The second thing it does is stating that monk weapons are shortswords and any simple melee weapon that doesn't have either the heavy property or the two-handed property.

That's everything that paragraph states. It simply tells you that monks fight unarmed or with monk weapons and then defines monk weapons as either shortswords or any simple melee weapon that isn't heavy or requires two-hands to use.

Simple melee weapon with heavy or two-handed property = not monk weapons.

Simple melee weapon without heavy or two-handed property = monk weapons.

So what about the versatile property? A weapon with that property just gives you the option to use two hands during an attack for greater damage. However, in this case whether a weapon has the versatile property or not has no impact on the RAW as it only purpose in this case is to define what a monk weapon is, which is a shortsword or any simple melee weapon without either the heavy property or the two-handed property. Any other properties such as light, finesse, reach or versatile has in this case no bearing on the rules. Nor does how you use said weapon.