Top seed Medvedev, who won just three games against Nadal in the final in 2019, the last time the tournament was played, prevailed 4-6 6-3 6-4 in his opener against Alexander Bublik.
The man Nadal beat to win the title the previous year, Tsitsipas, failed to convert on five match points in an epic second-set tiebreak but recovered to down Ugo Humbert 6-3 6-7 (13-15) 6-1.
Nadal pulled out ahead of a scheduled match against Lloyd Harris, who beat him last week in Washington as the 20-time grand slam winner struggled with a foot injury.
Countryman Feliciano Lopez replaces Nadal in the draw and will face Harris on Wednesday.
Elsewhere Tuesday, sixth seed Casper Ruud needed more than two hours to put away Marin Cilic 6-3 3-6 6-3, while 14th seed Grigor Dimitrov suffered a quick 6-3 6-4 exit against big-serving American Reilly Opelka.
A pair of unseeded veterans advanced, with 2016 finalist Kei Nishikori a 6-4 6-7 (5-7) 6-2 winner over Miomir Kecmanovic and John Isner defeating Alejandro Davidovich Fokini 6-4 6-1 in just over an hour.
Two days after appearing in his first ATP Tour final at the Citi Open, Mackenzie McDonald fell 6-3 6-4 to Benoit Paire in his Toronto opener.
In other first-round matches, Karen Khachanov beat Cameron Norrie 6-4 5-7 6-4, Frances Tiafoe downed Yoshihito Nishioka 6-4 6-3, Dusan Lajovic handled Emil Ruusuvuori 3-6 6-3 6-3 and Nikoloz Basilashvili defeated Jenson Brooksby 2-6 6-0 6-4.