Warner will be re-assessed next week ahead of the four-Test tour of India starting February 22, Cricket Australia said.
Usman Khawaja has been brought into the one-day squad to replace Warner who could still play in the final three ODIs.
"David Warner was struck on the left thumb by Mitchell Johnson in the nets," said CA physiotherapist Kevin Sims.
"While the fracture is very minor, it is not worth taking any risks. The injury will keep him out of at least the next two... matches against West Indies."
Australia skipper Michael Clarke, who missed the final ODI against Sri Lanka last week due to an ankle injury, is expected to play on Friday.
Warner, averaging 44.5 from 15 Tests, was still expected to be in the squad for India due to be announced later Thursday, but a batsman might be put on standby.
Fit-again all-rounder Shane Watson would be favourite to return to his favoured opening slot if Warner fails to recover in time.