US Open runner-up Medvedev, chasing his fourth ATP Tour title in a breakthrough 2019 season, will meet Germany's Alexander Zverev in Sunday's final.
The three-time Grand Slam winner says he no longer has pain in his hip and it is now a matter of building up fitness so he can play several matches in a row at tournaments.
Williams will face Canadian Bianca Andreescu for the US Open title as the American sets herself up for another attempt to clinch a record-equaling 24th Grand Slam singles title.
Champion Novak Djokovic elected to retire in the third set due to a left shoulder injury. He was facing off Stan Wawrinka who was leading 6-4, 7-5, 2-1.
Serena Williams and Maria Sharapova will open the evening session at Arthur Ashe Stadium with their first-ever NY meeting in a match that may just have the feel of a championship final.
Eighth-seeded Khachanov rallied for a 6-7 (3/7), 7-6 (7/4) 6-2 victory over the volatile Australian, but the final set was close to farce, with Kyrgios engaged in a running dialogue with Irish chair umpire Fergus Murphy.
Simona Halep won the Wimbledon final after two sets (6:2, 6:2) against Serena Williams. It is her second Grand Slam victory and the first in Wimbledon.
Federer and Nadal each won their quarter-finals to set up an epic semi-final between the two, whilst world number 1 Djokovic will face Bautista Agut for a place in the final.
Seven-time champion Serena Williams wobbled several times before overcoming Alison Riske on her way to the semi-final.
Tsitsipas had never won a set in his previous three meetings against claycourt specialist and 11-times French Open champion Nadal, but the Greek drew first blood by saving a break point to hold in the 10th game and take the first set.
Simona Halep, world number two, says she will withdraw from the Porsche Tennis Grand Prix in Stuttgart due to an ongoing hip injury in the Fed Cup over the weekend.
Rafael Nadal showed no signs of rust as he raced into a 5-0 lead against fellow Spaniard Roberto Bautista Agut in the first set on Court Rainier III, defending his Monte Carlo Masters title with 6-1, 6-1.
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