You can do it as follows:
SimpleDateFormat getTheCurrentLocaleDateTimeFormatString() {
    return (SimpleDateFormat) DateFormat.getDateInstance(DateFormat.FULL, Locale.getDefault());
}
A test program:
import java.text.DateFormat;
import java.text.SimpleDateFormat;
import java.util.Calendar;
import java.util.Locale;
public class Main {
    public static void main(String[] args) {
        Calendar calendar = Calendar.getInstance();
        SimpleDateFormat dateFormat = getTheCurrentLocaleDateTimeFormatString();
        String localeTimeString = dateFormat.format(calendar.getTimeInMillis());
        System.out.println(localeTimeString);
    }
    static SimpleDateFormat getTheCurrentLocaleDateTimeFormatString() {
        return (SimpleDateFormat) DateFormat.getDateInstance(DateFormat.FULL, Locale.getDefault());
    }
}
Output:
Sunday, 12 January 2020
[Update] 
Posting the following code based on your comment:
import java.text.DateFormat;
import java.text.SimpleDateFormat;
import java.util.Calendar;
import java.util.Locale;
public class Main {
    public static void main(String[] args) {
        Calendar calendar=Calendar.getInstance();
        String formatString=getTheCurrentLocaleDateTimeFormatString();
        System.out.println(formatString);
        SimpleDateFormat dateFormat = new SimpleDateFormat(formatString);
        String localeTimeString= dateFormat.format(calendar.getTimeInMillis());
        System.out.println(localeTimeString);
    }
    static String getTheCurrentLocaleDateTimeFormatString() {
        return ((SimpleDateFormat) DateFormat.getDateInstance(DateFormat.FULL, Locale.getDefault())).toLocalizedPattern();
    }
}
Output:
EEEE, d MMMM y
Sunday, 12 January 2020
[Another update] 
Posting the following code based on your another comment:
import java.text.DateFormat;
import java.text.SimpleDateFormat;
import java.util.Calendar;
import java.util.Locale;
public class Main {
    public static void main(String[] args) {
        Calendar calendar = Calendar.getInstance();
        String formatString = getTheCurrentLocaleDateTimeFormatString();
        System.out.println(formatString);
        SimpleDateFormat dateFormat = new SimpleDateFormat(formatString);
        String localeTimeString = dateFormat.format(calendar.getTimeInMillis());
        System.out.println(localeTimeString);
    }
    static String getTheCurrentLocaleDateTimeFormatString() {
        return ((SimpleDateFormat) DateFormat.getDateTimeInstance(DateFormat.FULL, DateFormat.FULL,
                Locale.getDefault())).toLocalizedPattern();
    }
}
Output:
EEEE, d MMMM y 'at' HH:mm:ss zzzz
Sunday, 12 January 2020 at 20:28:05 Greenwich Mean Time