How to use Firebase API on Android without the Google Play Service dependency?

I think you can't separate the two as Firebase is part of Google Play Services as mentioned in this post:

Firebase is part of Google Play Services. It is not really a separate SDK. However! Using Google Play Services as a whole is discouraged, to say the least. Developers are encouraged to only pick and choose the play services packages they need. In doing so, they “lose” Firebase capabilities. In order to get them back, you need to specify which Firebase packages you require, just like the other play services packages.

The Firebase SDK for Android is based on Google Play Services.
You can't avoid or skip the dependencies.

If you want to use the REST API you can check the official guide, but only a few services are available.

Here's an example of how to use Firebase Authentication on a device without Google Play services:

That will enable you to use Cloud Firestore, Realtime Database, and Cloud Storage on devices without Google Play services as well.