Radukanu was facing Victoria Golubic of Switzerland at the WTA 250 tournament in Nottingham when she felt “something pulled” on her body.
But the US Open champion said she expects to return to the court later on grass court.
“It was a shame to go out this week with a side injury and unfortunately (I) will no longer be able to play in Birmingham,” Radukan said in a statement released by the Lawn Tennis Association.
“I look forward to returning to the playing field soon to enjoy the rest of the grass season.”
Radukanou, 19, reached the fourth round of last year’s Wimbledon in her Grand Slam debut.
She then won the US Open in September, but has had form and fitness issues since her breakthrough last year and is currently without a coach.
On Tuesday, Radukanu said she had no idea if she could compete in Wimbledon, which starts on June 27.