Note that this answer was written in 2014. Since then, most night trains in western Europe have been discontinued, so the connections mentioned in this answer may not be available. As of 2017, it's likely that there is no longer any viable train option that allows you to travel at night, but I haven't checked what services remain in operation and what their schedules are.
Other answers show that with daytime trains you can just barely make it in one day if you drive the last leg from Valencia, or you can have a layover in Paris or Barcelona. The former Ellipsos train-hotel service between France and Spain was discontinued in December 2013 when the high-speed train line was opened.
However it is still possible to leave Antwerp in the late afternoon, cross France by a night train, and make your way to Valencia or Alicante where you can catch a regional train or bus or drive to Calp and arrive in the evening. This is rather long (more than 24 hours), but you save a hotel night. Your night train options are:
- Paris–Port-Bou, departing Paris-Austerlitz at 21:54 and arriving at 8:22. Antwerp–Paris takes 2 hours by hourly high-speed train, allow 1 hour to cross Paris comfortably.
From Port-Bou, you need to take a slow regional train, taking 2½ hours to reach Barcelona. From Barcelona, express trains take you to Valencia in 4 hours or to Alicante in 6 hours; faster Euromed trains take about 1 hour less.
For the return journey, the train departs Cerbère at 19:40 in the high summer and 20:16 starting in September.
- Paris–Latour-de-Carol, departing Paris-Austerlitz at 21:57 and arriving at 7:51. From Latour-de-Carol, a regional train takes you to Barcelona in about 3 hours, so you'll arrive in Barcelona at roughly the same time as with the Port-Bou train.
For the return journey, the train departs Latour-de-Carol at 20:20.
- Luxembourg–Port-Bou, departing Luxembourg at 19:28 and arriving at 9:17. From Antwerp to Luxembourg is a 4-hour hourly connection via Brussels, which makes the trip quite a bit slower than via Paris. Beware that this train runs only twice a week in winter.
For the return journey, the train departs Cerbère at 20:24.
From Valencia or Alicante to Calpe, you can take a TRAM train
or an ALSA bus. I can't seem to find a train schedule online. There is a direct bus from Valencia to Calpe taking 3 to 4 hours, with 9 services during summer weekdays; note that the last bus leaves Valencia at 18:00. From Alicante the bus takes 1½ to 2 hours and the last bus also leaves at 18:00.
I haven't researched bus options to cross France. They'll probably be slower but cheaper than the train.