De la Fuente and partner Cheryl Burke were performing a samba when the injury occurred. De la Fuente dropped Burke during the performance, unable to support her weight on a dip.
During Tuesday's broadcast, de la Fuente said he will need surgery but wants to delay the operation and not pull out of the competition.
"I would like to be there and not give up," he said before learning he was among the contestants voted to remain in the competition.
Actress Shannon Elizabeth was voted off the show Tuesday.
This was the seventh week of competition and the first in which contestants were required to learn two dance routines instead of one. It was the first time in the history of the show someone was injured during a live performance. Last season, Marie Osmond fainted.