No, except where a trait or feature explicitly allows it.
The question you linked addresses the definition of a natural weapon: they are "natural" body parts that function as weapons. As Rubiksmoose's answer to that question cites, they are described in the intro to the Monster Manual (p. 10-11) or the corresponding portion of the basic rules:
The most common actions that a monster will take in combat are melee and ranged attacks. These can be spell attacks or weapon attack, where the "weapon" might be a manufactured item or a natural weapon such as a claw or tail spike.
Unarmed strikes are more readily defined in a player-facing document, the PHB or basic rules, specifically the section of the combat rules on melee combat:
Instead of using a weapon to make a melee weapon attack, you can use an unarmed strike: a punch, kick, head-butt, or similar forceful blow (none of which count as weapons). On a hit, an unarmed strike deals bludgeoning damage equal to 1 + your Strength modifier. You are proficient with your unarmed strikes.
As you can see, unarmed strikes are definitively not weapons. Whether natural weapons count for features that require a weapon seems to be a more controversial matter (though answers lean towards yes in most cases).
Given just those definitions, it might seem like there could be some overlap between natural weapons and unarmed strikes. Personally, I think natural weapons are generally defined in a way that those body parts that are natural weapons clearly function like weapons, whereas the examples given for unarmed strikes generally indicate the use of a mundane body part in a manner it's not usually suited for.
In the rules
The rulebooks don't seem to explicitly refute the possibility of natural weapons being used for unarmed strikes - but the rules also don't state as a general rule that natural weapons can be used for unarmed strikes.
For every playable race that does have natural weapons, it is explicitly stated that they can be used for unarmed strikes, suggesting that this is an exception to the general rule; otherwise it'd be very redundant.
For instance, lizardfolk have the Bite trait (VGtM, p. 113):
Your fanged maw is a natural weapon, which you can use to make unarmed strikes. If you hit with it, you deal piercing damage equal to 1d6 + your Strength modifier, instead of the bludgeoning damage normal for an unarmed strike.
Tabaxi have the Cat's Claws trait (VGtM, p. 115):
Because of your claws, you have a climbing speed of 20 feet. In addition, your claws are natural weapons, which you can use to make unarmed strikes. If you hit with them, you deal slashing damage equal to 1d4 + your Strength modifier, instead of the bludgeoning damage normal for an unarmed strike.
Centaurs have the Hooves trait (GGtR, p. 16):
Your hooves are natural melee weapons, which you can use to make unarmed strikes. If you hit with them, you deal bludgeoning damage equal to 1d4 + your Strength modifier, instead of the bludgeoning damage normal for an unarmed strike.
Minotaurs have the Horns trait (GGtR, p. 19):
Your horns are natural melee weapons, which you can use to make unarmed strikes. If you hit with them, you deal piercing damage equal to 1d6 + your Strength modifier, instead of the bludgeoning damage normal for an unarmed strike.
And at 5th level, Simic Hybrids have the Grappling Appendages option for their Animal Enhancement trait (GGTR, p. 20):
You have two special appendages growing alongside your arms. Choose whether they’re both claws or tentacles. As an action, you can use one of them to try to grapple a creature. Each one is also a natural weapon, which you can use to make an unarmed strike. [...]
As you can see, every single time a playable race has a natural weapon, the rules clearly state that they can be used for unarmed strikes. If there were a general rule that natural weapons could be used for unarmed strikes, this would be a lot of needless repetition. The fact that it is repeated in this way every single time strongly indicates that the general rule is that natural weapons aren't used to make unarmed strikes, and that these racial traits define an exception to that rule.
There is no official ruling on this issue given in the Sage Advice Compendium as of 2019. Unofficially, rules designer Jeremy Crawford has clearly stated many times on Twitter that natural weapons aren't used for unarmed strikes by default.
For instance, in this tweet:
Ok, Kung-fu panda idea. Moon Druid 2/monk 1. Does a beast's natural 'melee weapon attacks' count as 'unarmed strikes'?
A natural weapon (a claw, horn, bite, etc.) is not an unarmed strike.
A second user interjected, and Crawford responded:
Then explain the following: Alter Form giving you natural weapons that alter unarmed strikes, and Aarakorcas.
Those are exceptional abilities that change the nature of a character's unarmed strikes.
Crawford reiterated this again a few years later:
Am I right that attacks with natural weapons are still considered unarmed strikes rather than attacks with a weapon? E.g. for greenflame blade, silvered fists and similar.
An attack with a natural weapon is not an unarmed strike. An unarmed strike follows the unarmed strike rules in the Player's Handbook, no matter what type of creature is attacking. Some exceptional natural weapons, such as tabaxi claws, can be used for unarmed strikes.
Is a bite an unarmed strike? Do teeth count as natural weapons? Are we starting to go down a rabbit hole? Is the white knight talking backwards?
No (unless you have a special feature). Sometimes (depends on the creature). No. No.
He has consistently stated that attacks with natural weapons are not unarmed strikes, except where the rules clearly state an exception. This is consistent with the rules references above.