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).
Related: Are weapons with the "Thrown" property considered a "Ranged Weapon" for the purpose of sharpshooter?
Somewhat related: Does a Monk's Martial Arts die replace all of a magic weapon's damage, or only the die portion of it?
Best Answer
Replacing improvised weapon's 1d4 with Martial Arts
While the damage die for monk weapons (and unarmed strikes) starts out as a d4, which also happens to be the die used for improvised weapons, this does not mean that a monk can treat improvised weapons as monk weapons.
The full description of damage for monk weapons makes it pretty clear that the "d4" is just the lowest level of the scaling die as indicated on the leveling table because it goes on to say (though you don't cite it):
You have it backwards: monk weapons use a d4 (that scales) because they are monk weapons. They are not monk weapons because they use a d4 (that scales).
Improvised weapons vs Monk weapons
Improvised weapons are not included in the list of weapons that may be treated as monk weapons. If a monk throws a monk weapon that lacks the thrown property, it is considered an improvised weapon which deals d4 damage (which is, coincidentally, the same die as the lowest level martial arts die).
Under the section header for improvised weapons, we see:
You express some misgivings about interpreting this to mean that such a weapon would be considered an improvised weapon because the rules say "it also deals 1d4 damage" instead of "it is also considered an improvised weapon."
The fact that this info appears under the section for improvised weapons and not the section about weapons in general makes it clear that the d4 damage die is the result of the weapon's improvised usage. It also serves to inform us that the attacker, in addition to not benefiting from their proficiency bonus, uses a d4 instead of whatever the weapon's die (or martial arts die, in the case of a monk) would normally be.
A note on your second related question
To clarify/correct your remark on the second related question where you say that the question/answer:
A versatile weapon wielded with two hands is not a two-handed weapon from the standpoint of the rules and therefore qualifies as a monk weapon regardless of how many hands are used to wield it.
Weapon types (1H, 2H, versatile) dictate how they may be wielded, not the other way around. That is, using a versatile weapon with two hands does not make it a 2H weapon any more than wielding a glaive with 1H makes the glaive a 1H weapon.
So to bring it all together
If a monk throws a monk weapon that lacks the thrown property (and is not deemed ergonomically identical to a monk weapon that does possess the thrown feature†), it is considered an improvised weapon and the attack does not benefit from the monk's proficiency bonus and deals 1d4+stength modifier damage (period).
Martial arts does not apply to the d4 of this ranged weapon attack because martial arts can't apply to improvised weapons.
For completeness, if the monk were to take the Tavern brawler feat, which grants the character proficiency with improvised weapons, the monk would be able to add her proficiency bonus to the attack but not her martial arts. It would still be an unscalable 1d4 because proficiency and status as a monk weapon are independent things.
†I'm not sure what weapon could be considered similar to a monk weapon only when thrown. Perhaps a shuriken? You can't really stab with a shuriken like you could with a dagger but they might be thrown similarly, depending on the size/design.