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.
However, natural weapons are considered weapons (though a separate category from simple and martial weapons), as reiterated later in the answer - though the designers did not always consider natural weapons when designing every game feature that references a "weapon" (so some elements of the game were designed with the intent of working with natural weapons, while others were not).
Based on these definitions, we can begin to assess whether there is overlap between natural weapons and unarmed strikes. Natural weapons seem to be defined in a way that those body parts that are natural weapons clearly function as weapons, whereas the examples given for unarmed strikes generally indicate the use of a mundane body part whose default/sole function is not as a weapon.
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 almost 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, almost every single time that 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. It's not proof, but it is a strong argument in favor of this interpretation.
The one exception to this trend is the dhampir lineage from Van Richten's Guide to Ravenloft. The description of the dhampir's Vampiric Bite trait says, in part (emphasis mine):
Your fanged bite is a natural weapon, which counts as a simple melee weapon with which you are proficient. You add your Constitution modifier, instead of your Strength modifier, to the attack and damage rolls when you attack with this bite. It deals 1d4 piercing damage on a hit. [...]
Here, the Vampiric Bite trait specifies that the bite natural weapon is considered a simple melee weapon, rather than being used to make unarmed strikes. Since the trait does not state that the bite can be used to make unarmed strikes, it can not. This sets it apart from all other playable races so far, and further reinforces that being a natural weapon doesn't automatically imply the ability to make unarmed strikes with them.
Official ruling in the Sage Advice Compendium
As of October 2020, the Sage Advice Compendium explicitly distinguishes natural weapons from unarmed strikes as part of its answer to a related question:
Are natural weapons considered weapons?
Things designated as weapons by the rules, including natural weapons,
are indeed weapons. In contrast, unarmed strikes are not weapons. They
are something you do with an unarmed part of your body.
As you can see, the SAC clearly indicates that natural weapons are weapons, and that unarmed strikes are not. This makes it clear that natural weapons and unarmed strikes are different things by default.
As noted above, a number of playable races' racial traits do explicitly make an exception to this, allowing those specific natural weapons to make unarmed strikes (even though natural weapons can't normally be used for unarmed strikes in general).
Unofficial rulings on Twitter
Unofficially, rules designer Jeremy Crawford has clearly stated many times on Twitter that natural weapons aren't used for unarmed strikes by default (e.g. in this pair of tweets in July 2016, and this one in January 2018, and this one from the following month).
He has consistently stated that attacks with natural weapons are not unarmed strikes, except where the rules clearly state an exception. This unofficial ruling is consistent with the rules references above, as well as with the official ruling in the Sage Advice Compendium.