Learn German: Er hat sich das Gesicht gewaschen.

In English, we use possessive words (my, his, her) for body parts. In German, it is much more common to use a dative reflexive pronoun (sich) to show who the body part belongs to, and then just use a definite article (das) for the body part itself. The German logic is essentially 'He washed the face for himself'.

Is sich in the accusative or dative case here?

It is in the dative case. When you wash your whole self, you use the accusative case. But because there is a specific direct object being washed here (das Gesicht), the reflexive pronoun shifts to the dative case. For the third person (er, sie, es), the dative and accusative forms are both sich, so it happens to look exactly the same!

If I wanted to say 'I washed my face', how would the sentence change?

You would need the first-person dative reflexive pronoun, which is mir. The sentence would become: Ich habe mir das Gesicht gewaschen. If you used the accusative mich here, it would sound wrong to a German speaker because of the direct object (das Gesicht).

Why is the past participle gewaschen instead of gewascht?