The American 16th seed had lost the first set on a tie-break to France's Chardy but, after a promising start to the second set, suffered back spasms for which he received treatment.
Roddick was broken twice after returning to the court, clearly struggling as Chardy triumphed 7-6(4) 6-3 and set up a second-round clash with Russia's Denis Istomin.
Recent wins at Atlanta and Eastbourne have boded well for the veteran Roddick, but now he has to decide whether to play next week or protect his body for the US Open.
"I don't know yet. You know, we're going to have to figure this thing out. I don't have all the answers for it yet," Roddick said.
"I felt fine early on, and then I had one lunge forehand and, we have all had it, your back kind of goes out or spasms a little bit. It got progressively worse.
"In a perfect world I'd love to play next week. I feel like I'd benefit from that. That's what we're going to try to do," he said, adding he would evaluate the situation in three days.
American 10th seed Fish easily beat Feliciano Lopez 6-2 6-3 on Centre Court. Fish was evenly matched by the Spaniard for the first half hour, but Lopez subsequently played poorly, double-faulting away the first set and putting up a token resistance in the second.
In another first-round match on Tuesday, David Nalbandian was beaten 6-7(0) 7-6(4) 6-3 by Germany's Tommy Haas.
In a gruelling three-hour match, Haas started well but collapsed after taking a 5-2 lead in the first set, losing 16 from 17 points at one point as Nalbandian won the tie-break to love.
But the German worked his way back into the match, winning a second-set tie-break after saving match point, and he capitalised on a trademark loss of temper by the Argentine, who blew his top with the umpire midway through the final set after incorrectly challenging a line call.
Haas is rewarded with a second-round match against world number nine and Olympic bronze medallist Juan Martin del Potro.
Cypriot star Marcos Baghdatis made light work of Alejandro Falla in a one-sided 6-4 6-1 result after the Colombian came in as a last-minute replacement for 11th seed Gilles Simon, who withdrew with a shoulder injury hours before the match.
Australia's highly-rated youngster Bernard Tomic was the first man into the third round as he eased to a 6-4 6-3 win over American Brian Baker in their second-round match.