Nadal blew a 5-2 lead in the third set in losing 6-3, 3-6, 7-5, his first defeat to Verdasco in 14 matches.
The third-round loss was Nadal's earliest exit in a clay-court tournament since he fell to Olivier Mutis in the second round in Palermo, Italy, in 2004.
Defending champion Djokovic and Roger Federer, meanwhile, eased into the quarterfinals with straight-set wins.
Nadal blamed his first loss on clay in almost a year on the blue clay, which players have said was slippery, and sent an ultimatum to organizers, threatening to strike the tournament from his calendar if the traditional red surface was not restored.
"Being able to move is very important for me and if I can't move well, I can't hit the ball well either," said Nadal, the second seed and two-time Madrid champion. "If things don't change, this will be one less tournament on the calendar for me.
"This surface destabilizes the game. It is a completely different game and I don't want to take risks."
Verdasco, who became only the seventh player to beat Nadal on clay in eight years, burst into tears on the court in front of his hometown fans upon sealing the upset with a forehand winner.
"After losing so many times against honestly the best player on clay ever, to beat him on clay is the maximum," said an emotional Verdasco. "I don't have words. I am happy for the win, although it is difficult to hold myself together now. I need to calm down, rest and get ready for the next match."
Verdasco will play Tomas Berdych next after the Czech beat Gael Monfils 6-1, 6-1.
Djokovic had an easier time defeating Stanislas Wawrinka 7-6 (5), 6-4, but said he would also boycott the tournament if it didn't go back to the traditional red-clay surface.
"They are claiming that the court is exactly the same as red clay, which is not true because there is a big difference," said the defending champion. "You are tripping, slipping all the time, sliding. The winner will be the one who doesn't get hurt by the end of the week."
"It's a new experience, and the way it looks this year, hopefully the last experience."
Djokovic let slip five set points before finally taking the first set in the tiebreaker and then broke his Swiss challenger twice to decide the match in an hour and 45 minutes.
The world's top-ranked player will next play fellow Serb Janko Tipsarevic, who beat Gilles Simon of France 7-6 (3), 5-7, 6-1.
The third-seeded Federer hit 10 aces and didn't concede a break against Gasquet en route to improving his record this season to 24 wins and three loses.
While not going as far as the sport's other top players, Federer said he understood their frustration.
"I understand (Nadal's) disappointment. He was against (blue clay) from the start and so was I," said Federer. "He never felt comfortable on the surface. It is a tough surface and that can only add to the anger even more."
The 16-time Grand Slam champion will face David Ferrer in the quarterfinals.
The fifth-seeded Ferrer saved three match points before smacking an unreachable shot into the corner to seal his hard-fought 7-6 (5), 3-6, 7-6 (8) victory over fellow Spaniard Nicolas Almagro.
Also, Alexandr Dolgopolov of the Ukraine ousted fourth-seeded Jo-Wilfried Tsonga 7-5, 3-6, 7-6 (2), and will meet Juan Martin del Potro after the Argentine beat Marin Cilic of Croatia, 6-2, 6-4.
On the women's side, Serena Williams overcame a sluggish start to beat Caroline Wozniacki 1-6, 6-3, 6-2 and reach the quarterfinals.
"I had a really slow start today. I don't know why, I was just a little sluggish and mentally I was fighting some demons maybe," said Williams, who extended her record to 10-0 on clay this season.
"I was just like, 'I gotta do better than this.'"
Williams will play second-seeded Maria Sharapova after her Czech opponent Lucie Safarova withdrew before their match due to illness.
Fourth-seeded Agnieszka Radwanska of Poland knocked off Roberta Vinci 7-6 (1), 6-4 and will play Varvara Lepchenko of the United States after she beat Anabel Medina Garrigues 6-1, 6-7 (3), 6-3.
Czech player Lucie Hradecka defeated Russia's Ekaterina Makarova 6-2, 7-6 (5) to advance to a meeting with fifth-seeded Sam Stosur.