Riske will meet Serena Williams in the last-eight after the seven-time victor dispatched Carla Suarez Navarro 6-2, 6-2.
Playing alongside Andy Murray, the 23-time Grand Slam victor also progressed through to the second round of the mixed doubles tournament.
The two-time Wimbledon champion advanced to his eighth straight Grand Slam quarterfinal with a 6-2, 6-2, 6-2 victory against Joao Sousa.
The match between Coco Gauff and Simona Halep will follow, while number 16 seed Johanna Konta and 2014 victor Petra Kvitova are the only two players still in the ladies' tournament to play on Centre Court today.
Also, No. 8 Elina Svitolina beat No. 24 Petra Martic 6-4, 6-2, and Zhang Shuai defeated Dayana Yastremska 6-4, 1-6, 6-2.
Barty was playing her first tournament as the No. 1-ranked player, and she started off by winning points with her serve against Riske.
Chief executive Richard Lewis called what Fognini said "a very unfortunate comment" but also "one of those heat-of-the-moment comments".
"I think she's doing everything great", Serena said, according to WTA Tennis.
More than 1,000 bombs fell in the area during World War II, destroying thousands of nearby homes, and 16 fell on the tournament grounds.
A star-filled cast that includes Serena Williams, 15-year-old sensation Coco Gauff and the men's Big Three of Roger Federer, Rafael Nadal and Novak Djokovic is on the schedule as Wimbledon resumes after its traditional day off.
"It is one of my favorite tournaments". Number six seed and 2014 victor Petra Kvitova takes on Britain's Johanna Konta on Centre Court, while Barbora Strycova faces number 21 seed Elise Mertens. That match on No. 1 Court follows seven-time Wimbledon champion Williams against Navarro.
Having dropped only one set each in their three rounds so far, the trio seem a safe bet to advance when they face fourth-round rivals who have never before made it to this stage at the All England Club.