날짜를 눌렀을 때는 'date' 모드, 시간을 선택했을 때는 'time' 모드를 사용하겠습니다. 컴포넌트를 다음과 같이 수정해보세요.

components/WriteHeader.js

import {useNavigation} from '@react-navigation/native';
import {format} from 'date-fns';
import {ko} from 'date-fns/locale';
import React, {useState} from 'react';
import {Pressable, StyleSheet, Text, View} from 'react-native';
import TransparentCircleButton from './TransparentCircleButton';
import DateTimePickerModal from 'react-native-modal-datetime-picker';

function WriteHeader({onSave, onAskRemove, isEditing, date, onChangeDate}) {
  const navigation = useNavigation();
  const onGoBack = () => {
    navigation.pop();
  };

  const [mode, setMode] = useState('date');
  const [visible, setVisible] = useState(false);

  const onPressDate = () => {
    setMode('date');
    setVisible(true);
  };

  const onPressTime = () => {
    setMode('time');
    setVisible(true);
  };

  const onConfirm = (selectedDate) => {
    setVisible(false);
    onChangeDate(selectedDate);
  };

  const onCancel = () => {
    setVisible(false);
  };

  return (
    <View style={styles.block}>
      (...)
      <View style={styles.center}>
        <Pressable onPress={onPressDate}>
          <Text>
            {format(new Date(date), 'PPP', {
              locale: ko,
            })}
          </Text>
        </Pressable>
        <View style={styles.separator} />
        <Pressable onPress={onPressTime}>
          <Text>{format(new Date(date), 'p', {locale: ko})}</Text>
        </Pressable>
      </View>
      <DateTimePickerModal
        isVisible={visible}
        mode={mode}
        onConfirm={onConfirm}
        onCancel={onCancel}
        date={date}
      />
    </View>
  );
}

(...)