The Spaniard, a double world champion, smiled and waved to reporters and fans as he left the hospital at around 1430 local time.
McLaren said in a statement that the 33-year-old would return to his family home in northern Spain for further rest and recuperation and would not be back in the car before the Australian season-opener on March 15.
"He will sit out this week’s final winter test, at which he was due to share driving duties with Jenson Button," the team said.
"Kevin Magnussen, McLaren-Honda’s test and reserve driver, will replace him."
Button will be first in the car at the Circuit de Catalunya on Thursday. Denmark's Magnussen raced for the team last season when they had Mercedes engines rather than the current Honda power unit.
Alonso has been in hospital since the crash on Sunday, with speculation running wild about what really happened when he hit the wall at the high speed Turn Three.
McLaren issued a detailed statement on Monday blaming gusting wind and ruling out any mechanical failure but that explanation, and Alonso's lengthy stay in hospital, has failed to silence all of the doubters.
The car did not appear significantly damaged after the impact and Ferrari's Sebastian Vettel, Alonso's replacement at the Italian team, told reporters the Spaniard's crash had appeared 'strange' to him.
McLaren, whose racing director Eric Boullier initially played down the crash as a "just one of those things that happens in testing", have rejected speculation that Alonso might have been rendered unconscious by an electrical fault.
The team, who have had a difficult pre-season with new partners Honda, said that rumour was "simply not true".
They have not said how fast he was going at the time of impact or given a precise reading of the G-forces he sustained, however.
(Writing by Alan Baldwin in London, editing by Ken Ferris and Justin Palmer)
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