Medvedev will go up against French tennis player Adrian Mannarino and Auger-Aliassime will take on The Netherlands' Tim van Rijthoven.
Nadal came into the French Open with little experience on clay, having only played five matches on the surface this year.
The King of Clay advanced to the quarter-final where he will face world no. 1 Novak Djokovic, who beat Diego Schwartzman in straight-sets.
The Norwegian Ruud came out as the victor after what was a thrilling contest that saw a brilliant tie-break in the very first set.
The Spaniard did not let his guard down even once in the final set against Soonwoo Kwon and recorded a comfortable 6-1, 2-6, 6-2 victory.
While the Canadian has not won a grand slam so far, he is the first player from the nation to reach the quarter-finals of the US Open.
At 17 years and one month, he is the second-youngest player to win multiple Challenger titles, and also is the youngest player to defend one.
Felix Auger-Aliassime is one of the hottest young prospects in the tennis world today. Read about the tennis racquet he uses here.
The third seed, who was competing in his ninth tour-level final, comfortably got past Stefanos Tsitsipas 6-4, 6-2 in 78 minutes.
Felix Auger-Aliassime is one of the hottest young prospects in tennis. Here we shall take a look at his earnings, prize money and other financial aspects.