The Spanish fourth seed, fresh from victory in a Madrid Masters triumph that saw him cruise past Serbian nemesis Novak Djokovic in the semis, will meet American Jack Sock in his third-round match later.
Djokovic got past Roberto Bautista Agut of Spain 6-4, 6-4.
Djokovic, beaten in the final by Britain's Andy Murray past year, will now face Juan Martin del Potro in the quarters after the unseeded Argentinian's 7-6 (7/4), 6-3 win over Japanese seventh seed Kei Nishikori. "Time TBD", said a statement on the tournament's Twitter account. "But I ll keep fighting".
Djokovic, a beaten finalist here previous year, held his serve throughout the opening set and earned the only break in the seventh game to ease into a convincing lead. "I was very happy with the performance".
"I feel like today's match was one of the better ones that I have played this clay-court season, but also in the past couple of months", Djokovic said.
In a shock to Maria Sharapova, the French tennis federation refused to grant the five-time Grand Slam victor and twice Roland Garros champion a wild card to either the main draw or qualifying. "I went out there with the belief that, if I played well and executed my game plan, that I could walk off that court with a win".
"I think I can play a little bit more aggressive than what I did tonight".
Downpour halts Djokovic-Del Potro quarter-final
Isner next faces another big server in sixth-seeded Marin Cilic , who dismissed ninth-seeded David Goffin 6-3, 6-4. Djokovic, a four-time champion in Rome, will meet either Pablo Carreno Busta or Roberto Bautista Agut next.
Nadal's quarterfinal opponent will be eighth-seeded Dominic Thiem, who saved three match points to beat American Sam Querrey 3-6, 6-3, 7-6 (7).
Men: Second round: David Goffin bt Fernando Verdasco 3-6, 6-3, 6-2; Tomas Berdych bt Carlos Berlocq 6-3, 6-4; John Isner bt Mayer 7-6 (4), 7-6 (4); Novak Djokovic bt Aljaz Bedene 7-6 (2), 6-2.
British fifth seed Johanna Konta reached the third round with a crushing 6-3, 6-0 win over Yulia Putintseva.
The 20-year-old will now take on fifth seed Milos Raonic, a 6-3, 6-2 victor against the 12th seeded Tomas Berdych.
Djokovic recently broke off his association with his coaching staff, playing in Madrid accompanied on the bench by his brother, Marko, and he said on Monday that he has not yet decided who will be his next coach.
Qualifier Daria Gavrilova beat seventh-seeded Svetlana Kuznetsova 2-6, 7-5, 6-4 and Kiki Bertens won 7-6 (3), 6-1 over Russia's Ekaterina Makarova.