As part of a three-way deal, the Grizzlies then traded Calderon to the Detroit Pistons for forwards Austin Daye and Tayshaun Prince.
"This is the type of deal we have been looking for to take our team another step forward," said Raptors president and general manager Bryan Colangelo. "Rudy Gay is a dynamic wing that elevates our talent level significantly.
"We want to thank Jose and Ed for their dedication and service to our organization. Both individuals are wonderful young men beyond just talented basketball players."
The Grizzlies signed Gay to a five-year, $82 million US maximum contract in July 2010. Gay is set to earn $16.5 million this season and $37 million for the next two years.
The 6-foot-8 small forward leads Memphis with 17.2 points per game and 36.7 minutes played. However, Gay is shooting a career-worst 40.8 per cent from the floor.
"Wow," Grizzlies point guard Mike Conley tweeted.
Calderon longest-serving Raptor
Calderon was the longest-serving member of the Raptors, having played his entire NBA career in Toronto. The six-foot-three Spaniard joined the team in the 2005-06 season and has played in 525 games, averaging 10.0 points and 7.2 assists per game.
"It's been my home for eight years," Calderon said in Atlanta, where the Raptors were taking on the Hawks, shortly before leaving the arena. "I've done everything possible for this team. It's tough. The fans have been with me since Day 1. It's tough."
Calderon was scheduled to become a free agent at the end of the season while Davis, who will make $3.1 million next season, has two more years left on his deal.
Toronto drafted Davis with the 13th overall pick in the 2010 NBA draft. Playing primarily in a reserve role, Davis has averaged 7.0 points and 6.9 rebounds over the last two seasons.
Calderon and Davis had both been starting for the Raptors, who were in 11th place before Wednesday's games, but still sit 2 ½ games behind Boston for the eighth seed.
"Hopefully this team is back to the playoffs as soon as possible," Calderon added.
The Raptors have been linked to Gay for some time now. The 26-year-old originally drafted eighth overall by the Houston Rockets in 2006 but was quickly traded to Memphis, along with Stromile Swift. The Rockets received forward Shane Battier in the deal.
"Wow that was 1 crazy trade today," Oklahoma City center Kendrick Perkins tweeted. "Are you serious Rudy Gay is right there under KD, Lebron, Kobe, and Melo. (hashtag)badtrade."
For Memphis, trading away Gay eases a luxury tax hit due next season, while concentrating the team around centre Marc Gasol and all-star forward Zach Randolph.
The Grizzlies had their best playoff run in 2011 when they knocked off then-No. 1 seed San Antonio before losing to Oklahoma City in seven games in the Western semifinals — all with Gay on the bench after needing season-ending shoulder surgery.
"We are excited to add three players who bring with them a tremendous amount of value to our team and have achieved incredible success on the pro, college and Olympic levels," Grizzlies GM general manager Chris Wallace said in a statement Wednesday night. "In these players, we welcome NBA Champion and Olympic gold medalist Tayshaun Prince, as well as up-and-coming athletic forwards Ed Davis, who won an NCAA title at North Carolina, and Austin Daye."