No, because of Monks in Wild Shape

If natural weapons can be used as unarmed strikes, a Druid 18/Monk 2 can do 25 damage¹ with +10 to hit as a bonus 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.

1. Mammoth form, 29 damage against prone enemies
2. 1d4+7 as Martial Arts