The 25-year-old was the highest-ranked seed due to play in Charleston.
Konta, 25, who became the first British woman to win the Miami Open with victory over Caroline Wozniacki on Saturday, was the highest-ranked seed due to play in Charleston.
"Charleston is a great tournament and I was really looking forward to taking part," Konta said in a statement.
"I was battling a slight shoulder injury and sickness during Miami which has taken hold since the end of the tournament."
With Konta out, Madison Keys becomes the highest ranked player in the tournament.
Venus Williams is also due to take part in the clay-court event.