You can get selected date in showPicker method from given code example :

showPicker = async (stateKey, options) => {
    try {
      var newState = {};
      const {action, year, month, day} = await DatePickerAndroid.open(options);
      if (action === DatePickerAndroid.dismissedAction) {
        newState[stateKey + 'Text'] = 'dismissed';
      } else {
        // <<<< Newly selected date >>>>
        var date = new Date(year, month, day);
        newState[stateKey + 'Text'] = date.toLocaleDateString();
        newState[stateKey + 'Date'] = date;
      }
      this.setState(newState);
    } catch ({code, message}) {
      console.warn(`Error in example '${stateKey}': `, message);
    }
  };
render() {
   return (
   <TouchableWithoutFeedback
    onPress={this.showPicker.bind(this, 'spinner', { date: this.state.presetDate })}>
     <View>
        <Text style={styles.text}>Date selector</Text>
     </View>
   </TouchableWithoutFeedback>
  )
}

As we can see the example code on documentation, create an async function, this function will handle promise data, and trigger this function to you component Text with onPress function or TexInput with onFocus function. example:

async _onMyDatePress(){
   try {
            const {action, year, month, day} = await DatePickerAndroid.open({           
                date: new Date()
            });

            if(action == DatePickerAndroid.dateSetAction){
                console.log(year + ' ' + month + ' ' + day);
            }

        } catch ({code, message}) {
            console.warn('Cannot open date picker', message);
        }   
}