The veteran had started to get into the match again after losing two straight sets when the accident occurred.
Roger Federer led 4-2 in the game in the fourth set when Mannarino slipped and twisted his leg badly.
He tried to continue playing, but was noticeably affected by the injury and chose to cancel after losing 2-2 in sets.
Federer is thus on to the second round of his favorite tournament.
The 20-time grand slam winner has won eight titles on the grass in London during his career - the most ever in the men's singles tournament.
Now he is looking to step up alone on 21 grand slams.
It is four years since the current world champion last won Wimbledon. Federer reached the final in 2019, but then fell to Novak Djokovic.