In a radio message which was not aired during Sunday's broadcast, Hamilton urged his team to pull him out of the race with 11 laps remaining.
After running off the road for a second time on a mistake-ridden afternoon at Hockenheim, Hamilton was 13th, and set to drop to last ahead of a sixth pit stop, when he made the call.
Mercedes, however, instructed Hamilton to continue.
"Okay, Lewis box, box for new soft tyres," said the Briton's race engineer Pete Bonnington on lap 53.
"Retire the car," Hamilton replied.
"Negative, Lewis, negative," said Bonnington. "There are always opportunities."
Hamilton would take the chequered flag in 11th. He was later promoted two places after both Alfa Romeo drivers were hit with time penalties for a technical infringement.
That would earn Hamilton two points to extend his championship lead over Valtteri Bottas to 41 points following the Finn's late crash on a miserable afternoon for Mercedes at their home event.
Hamilton had led for almost half the race before he crashed in the slippery conditions.
The 34-year-old will now spend the next few days at his Monaco home in the hope of recovering from a virus which hindered his preparations in Germany.
"Lewis wasn't healthy the whole weekend, but I think he did the most he could to get himself in an OK place for the race," said Mercedes team principal Toto Wolff.
"Many of us wouldn't have considered being in the car but he did. You can't physically be at your best when you've been ill for a few days. Having said that he tried to push through and that needs to be admired."
Hamilton will be back in action at this weekend's Hungarian Grand Prix, the final round before Formula One's summer break.
Meanwhile, the driver of a Renault team truck was taken to hospital following a motorway crash on Monday morning.
The vehicle was transporting machinery from Hockenheim to Budapest at the time of the accident.
A team statement confirmed the driver was extracted from the truck but did not suffer serious injury.
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