How to test a Method with an action delegate

I have the following call:

public IEnumerable<uint> DeleteAppointment(uint appointmentId)
    {
        return _dataContextProvider.GetContextRW().Run(db => DeleteAppointmentById(db, appointmentId));
    }

I want to test the private DeleteAppointmentById method but it isn't called of course. How can I call it?

The Run method:

void Run(Action<TDataAdapter> action)

How can it be void but still return the integer of the private function? I think I dont understand actions...