No hay nada en JDK para hacer esto.
static String[] suffixes =
// 0 1 2 3 4 5 6 7 8 9
{ "th", "st", "nd", "rd", "th", "th", "th", "th", "th", "th",
// 10 11 12 13 14 15 16 17 18 19
"th", "th", "th", "th", "th", "th", "th", "th", "th", "th",
// 20 21 22 23 24 25 26 27 28 29
"th", "st", "nd", "rd", "th", "th", "th", "th", "th", "th",
// 30 31
"th", "st" };
Date date = new Date();
SimpleDateFormat formatDayOfMonth = new SimpleDateFormat("d");
int day = Integer.parseInt(formatDateOfMonth.format(date));
String dayStr = day + suffixes[day];
O usando el calendario:
Calendar c = Calendar.getInstance();
c.setTime(date);
int day = c.get(Calendar.DAY_OF_MONTH);
String dayStr = day + suffixes[day];
Según los comentarios de @ thorbjørn-ravn-andersen, una tabla como esta puede ser útil al localizar:
static String[] suffixes =
{ "0th", "1st", "2nd", "3rd", "4th", "5th", "6th", "7th", "8th", "9th",
"10th", "11th", "12th", "13th", "14th", "15th", "16th", "17th", "18th", "19th",
"20th", "21st", "22nd", "23rd", "24th", "25th", "26th", "27th", "28th", "29th",
"30th", "31st" };