Joachim Low's side bowed out of the Euro 2016 on Thursday after a 2 0 defeat to hosts France, a result that the Manchester United midfielder addressed in an open letter on the DFB's official website.
"It's a real shame," he said. "Our dream has not been fulfilled. We didn't make it to the final, so Rio will not be followed by another magical night."
The world champions dominated long periods of the match, but it was only Les Bleus who could capitalise on incursions into the final third, with the excellent Antoine Griezmann scoring his second brace of the tournament.
"Unfortunately, we were missing something extremely important against France: a little bit of luck. Defeats are a part of the game though, even if they hurt," Schweinsteiger continued.
"We can accept our opponent's performance and sincerely and warmly congratulate the French team on progressing to the final.
"I am proud of what we have achieved. We have stayed true to playing our style of football. We have continued to grow as a team in the last seven weeks and this togetherness, which made us strong in Brazil two years ago, has done the same again here in France.
"Our fans are also a part of it. Wherever we were, Lille, Saint-Denis, Paris, Bordeaux or Marseille – thanks to you we had our twelfth man by our side."