The 34-year-old siblings beat French pair Michael Llodra and Jo-Wilfried Tsonga 6-4 7-6 (7/2) in the final at Wimbledon.
The 11-time grand slam winners said playing for their country brought a higher level of intensity to their game.
Bob Bryan said: "I can tell you there's no better feeling than this, winning for each other, for our team and for our country. It brought a different level of intensity to our tennis.
"To hear the national anthem and stand on the podium, we could shut our careers down today and be happy for the rest of our lives."
His brother said the win was a good end to what had been a "rough year", as it means they now have the career Golden Slam of all four majors and an Olympic title.
Mike Bryan said: "We saved it all for this one. We lost two grand slam finals this year, it's kind of been a little bit of a rough 12 months but we've got smiles on our faces now. There hasn't been a more special feeling than this.
"It's just something that you dream about. It's pretty cool to say you have the Golden Slam."
The Americans celebrated their victory by Mike leaping into his brother's arms.
Mike Bryan added: "Those moments, you don't plan them out. I just usually jump and he holds me up. That's the longest we've ever hugged."
Tsonga was upbeat after losing out on gold, but silver gave him plenty of satisfaction.
He said: "Of course we are disappointed about this final, but anyway it's something really good to get this medal. It's really something big and we will never forget this moment."
Julien Benneteau and Richard Gasquet, also from France, beat Spain's David Ferrer and Feliciano Lopez 7-6 (7/4) 6-2 to take bronze.