Thomson missed the 29 December start of the round-the-world race after an emergency appendix operation.
The 36-year-old was given the all-clear to rejoin Hugo Boss co-skipper Andy Meiklejohn but will now remain in the UK with partner Kate and son Oscar.
Stand-in skipper Wouter Verbraak will continue the race with Meiklejohn.
Thomson's operation, two days before the start in Barcelona, enabled him to be at his son's birth on 7 January, but though Oscar's condition has stabilised he will need close monitoring over the next few weeks.
"As a team we feel relieved that Oscar is being so well looked after and his condition is more stable," said Stewart Hosford, Hugo Boss team director.
"We fully understand and support Alex's decision not to join Hugo Boss as planned. We have every confidence that Andy and Wouter will do us proud."
Hampshire's Thomson made his mark in ocean racing at the age of 25 when he became the youngest skipper ever to win a round-the-world race - the Clipper Race in 1998/99.
In the first edition of the Barcelona World Race in 2007-8, Thomson and co-skipper Andrew Cape set a new monohull world speed record, covering 501.3 nautical miles (576 miles, 928km) in 24 hours in a previous version of his Hugo Boss yacht.
The pair went on to finish second of the nine competing boats behind Frenchman Jean-Pierre Dick and Ireland's Damian Foxall.
Later in 2008, Thomson was one of the favourites in the single-handed Vendee Globe non-stop around-the-world race - in which Dame Ellen MacArthur made her name by finishing second in 2001 - despite facing a race against time to make the start line.
His Hugo Boss yacht was struck by a French fishing vessel on the delivery trip to Les Sables d'Olonnes three weeks before the start and his shore crew worked around the clock to make repairs.
But just three days in, Thomson discovered the boat was leaking and he was forced to abandon the race.
On his debut in the Vendee Globe in 2004, Thomson was forced to retire early on with damage to his yacht.
His boat also lost its keel in the Southern Ocean during the 2006 Velux Five Oceans around-the-world race and he was rescued by fellow Briton Mike Golding.
The 25,000 nautical mile (46,300km) Barcelona World Race begins in Spain on 31 December and takes an easterly route around the world, passing to the south of the Cape of Good Hope, Australia and Cape Horn, with a diversion through the Cook Strait between the North and South Islands of New Zealand.
By BBC Sport