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Even without that text however, 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).

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, is to use 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 wherein you say that it:

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).

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 d4 because proficiency and status as a monk weapon are independent things.

Replacing improvised weapon's 1d4 with Martial Arts

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

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:

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).

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.

You express some misgivings about interpreting this to mean that such a weapon would be considered an improvised weapon b 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, is to use a d4 instead of whatever the weapon's die would normally be.

Weapon types (2H, versatile, 1H) 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 it 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).

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 d4 because proficiency and status as a monk weapon are independent things.

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, is to use a d4 instead of whatever the weapon's die (or martial arts die, in the case of a monk) would normally be  .

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).

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 d4 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.}

I think you're assuming/interpreting that, because the damage die for a Monk's unarmed strikes and monk weapons starts out as a d4, which also happens to be the die used for improvised weapons, it means that a monk can treat improvised weapons as monk weapons.

The full description of damage for monk weapons and unarmed strikes 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 can roll a d4 in place of the normal damage of your Unarmed Strike or monk weapon. This die changes as you gain monk levels, as shown in the Martial Arts column of the Monk table.

Even without that text however, 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 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:

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.

You express some misgivings about interpreting this to mean that such a weapon would be considered an improvised weapon b 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, is to use a d4 instead of whatever the weapon's die would normally be.

A note on your second related question wherein you say that it:

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

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 (2H, versatile, 1H) 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 it 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 d4 because proficiency and status as a monk weapon are independent things.

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 can roll a d4 in place of the normal damage of your Unarmed Strike or monk weapon. This die changes as you gain monk levels, as shown in the Martial Arts column of the Monk table.

Even without that text however, 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 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:

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.

You express some misgivings about interpreting this to mean that such a weapon would be considered an improvised weapon b 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, is to use a d4 instead of whatever the weapon's die would normally be.

A note on your second related question wherein you say that it:

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

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 (2H, versatile, 1H) 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 it 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 d4 because proficiency and status as a monk weapon are independent things.