Partnering with 29-year-old Salisbury, who was playing his first Olympics, the British duo beat second seeds Herbert and Mahut in the first round, getting off to a solid start.
The Brits looked on course to make the semi-finals, when the medal matches in the tennis event begin, but later lost their set and a break lead to their Croatian opponents, falling 6-4 6-7(2) 7-10.
After the match, the former world number one refused to commit to playing the Paris 2024 games and admitted he may never have a chance to fight for another medal again, currently owning two golds in singles and one silver in mixed doubles.
He said: “I don’t know about Paris. I’ve got to get the opportunity to play again.
“I’ve loved every minute of playing in the Olympics. I wished that today could have gone differently.”