The Spanish dream team lost their quarter-final in Paris 6-2, 6-4 to the US fourth seeds Austin Krajicek and Rajeev Ram.
Nadal and Alcaraz, dubbed "Nadalcaraz", have been one of the main talking points at the Paris Games tennis tournament, wowing the crowds in their first two matches.
But they came unstuck in the last eight against doubles specialists Krajicek and Ram, who both have Grand Slam-winning pedigree.
Nadal, a 14-time French Open champion, is immortalised with a statue at Roland Garros.
But the 38-year-old has slumped to 161 in the world after a succession of injuries and faces relentless questions over his future.
He said after his painful defeat to Novak Djokovic in the singles competition in Paris that he would make a decision on his future after the Olympics.
The Spaniard -- second on the all-time list in men's tennis with 22 Grand Slams -- walked off Court Philippe Chatrier on Wednesday applauding all four sides of the stadium.
He was later asked whether he had played at Roland Garros for the final time.
'Unforgettable feeling'
"Maybe, I don't know," he said. "If that's the last time, for me it's an unforgettable feeling and emotions.
"They give me the love and the support every single second that I have been on court.
"For me, it's super, super special to feel that, particularly in this place. I can't really thank enough all of them for giving me the feelings that I have inside myself and the emotions that they make me feel."
He also suggested he would probably not play at this year's US Open which he has won four times.
"I need some time but for me it looks difficult," he added.
Nadal, who won Olympic singles gold in Beijing in 2008 and doubles gold eight years later in Rio, said it had been a "great experience" to be part of the Spanish team at the Paris Games.
"For me personally it was disappointing to not bring back a medal for Spain but that's it," he said.
"I tried my best in every single minute that I was on court but it was not enough to achieve our goals."
And he said the Olympics had a special place in the sporting calendar.
"Honestly, the feeling and the emotions that you live at an Olympic Games are difficult to compare with other events because here you feel part of something bigger than only a personal team," he said.
"That's an amazing feeling. I feel so lucky that I have been able to win a gold in singles because in the end, it's the most difficult event to win because you have just a few chances in your career. That's the truth."
He added: "This time was not possible, but that's it. We tried and the Olympic Games, for me, without a doubt, is the most important event in the world of sport."
Alcaraz, a four-time Grand Slam winner at the age of just 21, is through to the men's singles quarter-finals, in which he is the second seed.
But he said teaming up with Nadal had been unforgettable.
"It was an unbelievable experience for me, something that I will never forget, that's for sure, playing on the same side of the net with Rafa, learning from him."
