One detective leading an inquiry into the incident said it made him "ashamed to be British".
Inspector Jeffrey Whitemore said the family was returning to their hotel from a day trip to the Robin Hood centre in Sherwood Forest when a youth jumped in front of their Mercedes car in the Bulwell area of Nottingham, forcing it to stop.
The family was then attacked by the gang, who had been drinking and were chanting racist slogans at them, referring to Germany's victory over England in the semi-final of the Euro 96 football tournament.
They attacked the driver, Dr Heinrich Wagner, 50, his 49-year-old wife, Erika Rust, and their sons, Hubertus, a 20-year-old student at Nottingham University, and Jurgen, 18.
"The father suffered a black eye. One of the young Germans was hit in the cheekbone. The mother was also assaulted, but to a lesser degree.
"They are all very shocked," said Inspector Whitemore.
The family received hospital treatment but were not detained.
Dr Wagner, who lives in Mellrichstadt, north-east of Frankfurt, spoke of his dismay after the attack. "It's the first time in our whole lives that something like this has happened. We did not expect this in a civilized country," he said.
"If we had gone to Bosnia we would have expected it, but never in England."
Inspector Whitemore said:"We are treating this as a racial incident. It started when the gang saw the German number plates on a large Mercedes.
"It was a very nasty attack and it makes you ashamed to be British," he said. The Wagner family was combining a holiday in Britain with a visit to their son at Nottingham University.
A youth was later questioned by police in connection with the incident.
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