As temperatures rocketed in the French capital, officials activated a heat protocol, allowing a 10-minute break between the second and third sets.
In the first match on Court Philippe-Chatrier, Vekic shocked Gauff 7-6 (9/7), 6-2 to reach the quarter-finals but the match was overshadowed by a lengthy row between the US Open champion and the chair umpire.
Gauff, the flag-bearer for the United States at the Olympic opening ceremony alongside LeBron James, made a fast start but squandered three set points before her Croatian opponent won the first-set tie-break.
The pair swapped breaks early in the second set but the match came to a standstill at the end of the sixth game as an emotional Gauff pleaded her case with the chair umpire.
At 30-40, Vekic hit a return deep to Gauff's forehand, which the American mishit into the net.
Vekic's shot was called out but the call was overruled by the chair umpire and the point was awarded to the Croat, giving her the break. Gauff argued her shot was compromised by the initial call.
"I have to advocate for myself all the time," said a weeping Gauff, 20. "I'm getting cheated in this game. You guys are not fair to me."
Vekic, ranked 21, briefly lost her composure, slipping to 0-40 on her own serve before recovering to hold, and she broke again to seal victory.
"It took me some time to adjust to the conditions because when we warmed up the roof was closed so it was completely different when we came out there," she said.
"The ball was flying a little bit more so it took me a couple of games but after that I think I played pretty well and it was a really tough match."
Next up on Philippe-Chatrier was defending men's champion Alexander Zverev, who was taking on 39th-ranked Czech Tomas Machac before women's world number one Iga Swiatek targets a 24th straight win on the clay courts of Paris.
Poland's Swiatek, who has not lost at Roland Garros since 2021, takes on China's Wang Xiyu in the evening session.
'Nadalacaraz'
Nadal, a 14-time French Open champion, was swept off court by old rival Novak Djokovic in straight sets in the pair's 60th meeting on Monday.
The 38-year-old, immortalised in a statue at Roland Garros, has been plagued by injuries in recent years and slumped to 161st in the world.
The former world number one said after his painful defeat to Djokovic that he would make a decision on his future after the Olympics.
But the Spaniard, who won singles gold at the 2008 Beijing Games and doubles gold in Rio in 2016, can still collect a medal in the doubles alongside current French Open and Wimbledon champion Carlos Alcaraz.
They take on Dutch pair Tallon Griekspoor and Wesley Koolhof.
Another player in the final chapter of a storied career is Britain's Andy Murray, who, like Nadal, has battled crippling injuries over the past few years, slumping to 117 in the world.
The British three-time Grand Slam champion pulled off an astonishing escape act with doubles partner Dan Evans in their opening match, saving five match points in the deciding tie-break against Kei Nishikori and Taro Daniel.
