The double world champion was concussed and airlifted to hospital
when his McLaren ran wide during winter testing at the Circuit de Catalunya -- home to the Spanish Grand Prix -- on Sunday.
"We are pleased to confirm that Fernando Alonso has now left hospital," McLaren said in a statement.
"He has returned to his family's home in Spain for further rest and recuperation."
Alonso gave a thumbs-up to photographers as he walked out of the Cataluna Hospital on Thursday and drove away with his girlfriend.
Kevin Magnussen, who effectively lost his McLaren seat to the returning Alonso, will share driving duties with Jenson Button at the four day test which begins on Thursday.
Alonso chose to leave Ferrari at the end of the 2014 season and rejoin McLaren, which is also renewing its technical partnership with Japanese engine manufacturer Honda for 2015.
The 33-year-old -- regarded by his peers as the best driver on the grid -- is chasing a third world crown but, so far, he has not had chance to gauge the possibilities of realizing that ambition with McLaren.
Early reliability problems meant Alonso had only completed 117 laps in the MP4-30 race car before his preseason was curtailed by the crash.
McLaren had explained that Alonso's "accident was caused by the unpredictably gusty winds at that part of the circuit at that time."
Turn Three, where the accident took place, is an area of the circuit where cross winds are known to affect the balance of the cars.
The team statement continued: "We can also disclose that no electrical discharge or irregularity of any kind occurred in the car's ERS [energy recovery] system either before, during or after the incident.
"That last point refutes the erroneous rumors that have spread recently to the effect that Fernando was rendered unconscious by an electrical fault. That is simply not true."
McLaren sporting director Eric Boullier had described the crash as "one of those things that happens during testing."
Alonso will be expected to return to driving duties for CNN sponsored McLaren in first practice for the season opening Australian Grand Prix on March 13.