NoGenerally no, because of Monks in Wild Shapebut there are exeptions
If all natural weapons cancould be used as unarmed strikes, a Druid 18/Monk 2 cancould do 25 damage1 with +10 to hit as a bonus actionbonus action. Twice per short rest, it can be 2 attacks per bonus action.
If these are separate things, the bonus attack only does about 9 damage2, with 13 to hit. Much more balanced.
I guess not allowing this was a conscious decision by the game designers. Of course, now they have to make every racial natural attack an exception, but those do not cause balance issues with 1d4 or 1d6 damage.
The exceptions are mentioned in the description
For example, the bite of Lizardfolk is a natural weapon (VGtM, p113):
Your fanged maw is a natural weapon, which you can use to make unarmed strikes.
- Mammoth form, 29 damage against prone enemies
- 1d4+7 as Martial Arts