Emma Laducane was injured four days before the French opening in a three-set defeat to Daniel Collins on Wednesday.
Britton went down 5-0 second when he called a medical timeout for treatment with a back issue after winning the first set.
While she finished the match, the 22-year-old won four more games against Collins, with the Americans winning 4-6, 6-1, 6-3 to reach the quarter-finals.
Collins has performed top seed Jessica Pegra or the world’s 30th Anna Kalinskaya over the past eight years.
Laducanu has been an impressive recent Clay – winning three matches at the Italian Open last week, breaking up Dahlia Casatokina to start Strasbourg, but now she may be sweating her fitness before the French Open.
The tournament begins on Sunday at Roland Garros and there will be a first round draw on Thursday.