The pair had raised expectations with two fine performances to make the last eight and led Croatian duo Marin Cilic and Ivan Dodig by a set and a break.
They had a point to make it 5-3 but Cilic and Dodig stepped up their level, broke the Murray serve and prevailed in a deciding tie-break to win 4-6 7-6 (2) 10-7.
It is they who are now guaranteed to play for a medal, and the disappointment for Murray and Salisbury was clear.
“It’s always tough when you lose matches but especially here, a chance of getting to the semi-finals and a chance of winning a medal,” said Salisbury. “And the situation we were in, we were a set and a break up and we were playing well, so it’s very tough.”
Murray is the only tennis player ever to win successive gold medals, and he also has a silver in the mixed doubles with Laura Robson from London.
Given his ongoing struggles to stay fit, which included pulling out of the singles here with a thigh strain, it seems highly unlikely he will compete at a fifth Games in Paris in 2024.
“I don’t know if I’ll get the opportunity to play again,” he said. “I’ve loved every minute of playing in the Olympics. I wish that today could have gone differently.
“I had another chance with Joe to win a medal, we were so close, and that’s what’s disappointing. I would like to have done some stuff differently in the match to try to help out more.”
Murray will now head home for more rest and recovery but is hopeful of being fit for the US Open, starting on August 30.
He said: “I’ll see how the injury heals. That’s what will dictate it. I’m not going to rush something if it’s not there.”