The version played included the first stanza, beginning "Deutschland, Deutschland uber alles," which was used as Nazi propaganda. It was dropped after World War Two.
"I thought it was the epitome of ignorance, and I've never felt more disrespected in my whole life, let alone in Fed Cup," Germany's Andrea Petkovic said, adding that she considered walking off court before the singles match against Alison Riske.
She later said her first comments had been somewhat emotional but that was because the incident happened before her game and she was left stunned.
"We were left shocked and did not know how to react," Petkovic said.
German team coach Barbara Ritter said the mistake was "an absolute scandal, a disrespectful incident and inexcusable".
German tennis federation chief Ulrich Klaus said his American counterpart had apologised for the mistake.
"Our American hosts at the Fed Cup opening in Hawaii made a mistake that should not happen," Klaus said in a statement.
"The fact that in the year 2017 a wrong anthem can be played that is associated with the horror of the past was for players and staff and the officials present both shocking and disturbing."
"The USTA through its president Katrina Adams has apologised officially in writing and in person and deeply regrets the blunder."
Riske won the opening rubber to give the U.S. a 1-0 lead. Bad weather prevented any further play on the opening day of the tie.
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