Meintjes received stitches to an elbow injury, and was due to be reassessed for any delayed signs of concussion before being allowed to continue on Sunday's stage 15 to Lagos de Covadonga.
The race had split on the descent of the Alto de La Mozqueta, and the Dimension Data leader had been riding to make up lost ground on the day's penultimate climb of the Alto de la Falla.
It was descending this mountain that he crashed, when his front wheel "washed out in a high-speed corner", according to his team's website.
"He crashed on a corner and rolled down an embankment," team doctor Jarrad van Zuydam explained.
"He has suffered some injuries, and has a big laceration to the left elbow which required stitches. He has various other lacerations and abrasions which are relatively minor and won't need much more then cleaning and dressing.
"There was obviously some concern when we saw the footage of the crash, because of the way Louis stumbled, and there were concerns of a potential head injury and concussion in particular," said Van Zuydam.
"Louis has been assessed thoroughly and he currently does not show any symptoms of a concussion. He will be reassessed in the morning [Sunday], and if he passes that test, he will undergo an exercise stress test to see if that induces any symptoms before we give any go-ahead for him to race."
Going into Saturday's stage, Meintjes was already well off the pace in 24th place overall, more than eight minutes behind race leader Jesus Herrada (Cofidis).
After battling on after his crash to finish the stage over 16 minutes down on stage 14 winner Simon Yates (Mitchelton-Scott), the 26-year-old fell only a few places to 32nd overall, but is now 22:40 behind new race leader Yates.
Assuming he can recover from his injuries, Meintjes will lead the South African team at the upcoming world championship road race in Innsbruck, Austria, on September 30.