The second-seeded American won the first set 6-4 against the 29-year-old from Luxembourg but sustained an apparent ankle injury in the fifth game of the second set.
He managed to serve out the game to take a 3-2 lead but then decided he was unable to continue.
Muller will face Matthew Ebden in the quarter-finals after the Australian rallied from a set down to overcome American James Blake.
Ebden secured the decisive break in the ninth game of the third set en route to a 6-7 (6/8) 6-4 6-4 win.
American top seed John Isner was also taken to three sets, but progressed thanks to a 4-6 6-3 6-4 triumph against Ruben Bemelmans.
World number 11 Isner, aiming for his first title in Atlanta after finishing runner-up to Fish for the past two years, hit 16 aces against the Belgian qualifier.
Next up for Isner will be 19-year-old compatriot Jack Sock, who progressed to his first ATP Tour quarter-final with a 7-6 (7/5) 7-5 (7/5) victory over fellow American wild card Steve Johnson.