The monk's martial arts ability states:
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.
I am specifically asking about a shortsword, which is a monk weapon by definition, but this question could apply to any monk weapon that does not have the thrown property.
The monk's martial arts ability also states:
You gain the following benefits while you are unarmed or wielding only monk Weapons and you aren't wearing armor or wielding a Shield... • You can roll a d4 in place of the normal damage of your Unarmed Strike or monk weapon.
I would be more comfortable if this said "You can roll the Martial Arts Damage Die shown in the Monk Level / Ability Progression Table" rather than "you can roll a d4", but the other questions I looked at in researching this took that as an assumption and no one challenged them, so...(Note: Rykara pointed out that the rules actually do indicate the table is to be used. Unfortunately the online source I was citing did not have that line from the PHB)
The rules for using an improvised weapon state...
If a character uses a ranged weapon to make a melee Attack, or throws a melee weapon that does not have the thrown property, it also deals 1d4 damage.
The rules for using an improvised weapon also state...
At the DM’s option, a character proficient with a weapon can use a similar object as if it were that weapon and use his or her proficiency bonus.
Unfortunately, that and the Tavern Brawler feat are as close as we get in RAW to saying that "If you use an improvised weapon to make an attack, you normally don't get to add your proficiency bonus to the attack." I think we can take it as well-accepted although not explicitly stated that using a melee weapon without the thrown property to make a thrown attack does not count as an attack with a weapon with which you are proficient.
Consider a monk wielding a shortsword and fitting all other requirements for using the martial arts ability.
The shortsword is a melee weapon that does not have the thrown property and has a base damage of d6.
If the monk throws the shortsword it would become an improvised weapon, and thus normally would deal d4 damage.
However, in this case does specific beat general and allow the monk to replace the d4 "normal damage" of an improvised monk weapon with the martial arts die damage, which would be d6 at 5th level?
Or, does the fact that the monk is throwing the shortsword as an improvised weapon disqualify it from being considered a monk weapon, in the same way that one cannot use proficiency for an improvised weapon attack because throwing a melee weapon without the thrown property is no longer a weapon with which one is proficient, and thus remove the ability to use the martial arts feature?
This question: Does Martial Arts Damage Apply to Ranged Attacks with Monk Weapons? affirms that monks get their martial arts die damage on ranged weapons, but the question only considers melee weapons that already possess the ranged property, not improvised weapons.
This question: Does using versatile weapons with 2 hands disqualify them as Monk weapons? affirms that monks get to use their martial arts feature on all monk weapons, even when they use them in ways that would disqualify them from being monk weapons (in this case, using a versatile weapon two-handed).