I need to change the font size of CupertinoDatePicker so it looks more like native one. The font is small compared to ios datepicker. 
For the height wrapping in a Container with height of MediaQuery.of(context).copyWith().size.height / 3 makes the height to be similar to native one.
You can change the font size of text in a Text Widget using style property. Create a TextStyle object with fontSize and specify this object as style for Text Widget.
Another way to do this, is to wrap the CupertinoDatePicker in CupertinoTheme.
CupertinoTheme(
    data: CupertinoThemeData(
        textTheme: CupertinoTextThemeData(
            dateTimePickerTextStyle: TextStyle(
                fontSize: 16,
            ),
        ),
     ),
     child: CupertinoDatePicker(
          ...
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