Leading drivers have threatened to withdraw from Sunday's Formula One race if tire problems persist, while teams were asked to stick to operating requirements.
Hamilton set the pace in practice, completing the fastest lap in 1 minute, 31.754 seconds. Hamilton is still seeking his first win since joining Mercedes this season.
Rosberg, winner of two of the last three races, was .219 seconds behind. Red Bull's Mark Webber was third and Force India's Adrian Sutil was fourth.
Championship leader Sebastian Vettel, still looking for his first win on home soil, was eighth fastest. His closet challenger, Fernando Alonso, failed to complete a lap in his Ferrari, apparently due to electrical problems.
The tire problems became an issue at last week's British GP, where five cars sustained blowouts, leading drivers to express concern for the safety of everyone at the track.
Pirelli wants all teams to remain within the prescribed pressure limits and to stop switching tires from left to right and vice versa.
The International Automobile Federation said on Friday it was the responsibility of each team to comply with the requirements.
Late Thursday, the Grand Prix Drivers' Association said the drivers would pull out if tire problems persist in Germany.
"The drivers have decided if similar problems should manifest themselves during the German GP, we shall immediately withdraw from the event, as this avoidable problem with the tires endangers again the lives of drivers, marshals and fans," The GPDA said.
This weekend, the belt on the rear tires will be made of Kevlar instead of steel in an effort to reduce the chance of punctures.
From the Hungarian GP at the end of this month, the tires will change again, using the 2012 construction with 2013 compounds.
In the paddock at the Nuerburgring, some drivers praised Pirelli for acting quickly to rectify the problem. But the GPDA, chaired by Pedro de la Rosa and with Vettel and Jenson Button as directors, met later and decided to take their own action in order to avoid a repeat of the scenes from Silverstone.
Not all divers are members of the GPDA - Kimi Raikkonnen of Lotus and Sutil are among non-members.