Honda employs more than 700 staff at the team's base in Brackley, Northamptonshire, as well as the Formula One drivers, Jenson Button and Rubens Barrichello.
Fukui said the team may be put up for sale, adding that Honda's top priority was to "protect its core business activities".
Al Jazeera's sports correspondent Joanna Gasiorowska called Honda's decision a critical development for the Formula One teams, though not really a surprise.
"We would love to be able to continue in Formula One but we're simply not able to in the current financial climate"
Honda chief executive
"Over the last couple of months, Formula One bosses have been warning all of the teams, saying they must come up with some cost-cutting measures to combat the economic crisis," she said.
She said that if the industry does not manage to cut costs soon enough, more teams may have to pull out.
Measures discussed are to develop more fuel-efficient engines or more radically, to standardise them.
Honda's departure from the multi-billion-dollar glamour world of Formula One comes amid increasingly gloomy times for the automobile industry.
In Washington on Thursday chief executives from the big three US automakers - GM, Ford and Chrysler - returned to congress to appeal again for a government lifeline they say is vital to keep one or possibly all of them from going under.
It will leave the sport with a depleted grid of just 18 cars if no buyer can be found before the 2009 season kicks off in March.
The move will also prompt fears that other major manufacturers, with their factory production suspended and thousands of staff laid off, could follow Honda's example.
Honda and fellow Japanese motor giant Toyota have been the big spenders in Formula One in recent years.
Honda announced on Thursday it was cutting hundreds of full time and temporary jobs at its plants in the UK and Japan because of plunging vehicle demand.
It has already reduced its annual production of consumer cars by more than 140,000 worldwide in response to plunging demand in the US and other key markets.