Hamilton, who fought his way through the field from 20th on the grid to finish third, damaged his front wing by touching Button’s car as he tried to pass his fellow Briton.
Home favourite Nico Rosberg capped a fortnight in which he wed, cheered Germany’s World Cup triumph and signed a new contract with his fourth win of the season, which extended his championship lead over Mercedes team-mate Hamilton to 14 points.
Hamilton might have overtaken Valtteri Bottas of Williams for second but for his damaged car, which he admitted was down to his own “bad judgement” – and Button agreed.
“I think the problem with Lewis is he expected me to let him past,” said Button, who finished eighth. “I don’t think I’m the only person he drove into today. It’s strange, when the car’s so much quicker you’d think he wouldn’t get into fights. Why would we let anyone through?”
Champion Sebastian Vettel was fourth and Red Bull colleague Daniel Ricciardo sixth, behind Ferrari’s Fernando Alonso, while Felipe Massa escaped unhurt after flipping his Williams on lap one.
Hamilton’s qualifying errors were compounded by a five-place grid penalty for changing his gearbox, but he minimised the damage to his title chances with a dogged drive.
1. Nico Rosberg, Mercedes 190
2. Lewis Hamilton, Mercedes 176
3. Daniel Ricciardo, Red Bull 106
4. Fernando Alonso, Ferrari 97
5. Valtteri Bottas, Williams 91