"I think he is out for the tri-series so the most important thing from my aspect is that he is right for the test series (in Sri Lanka)," Smith told Cricket Australia's website.
"I think that's about six weeks away and the medical staff and Dave are confident he will be fine for that first test match."
Warner was named man of the match on Saturday after completing his first ODI century outside Australia, a knock of 109 off 120 balls at the appropriately-named Warner Park.
He has also scored a half century in his three innings in the Caribbean and is averaging 82.50.
"It's tough missing guys like Dave. He's in the form of his life and batting beautifully and he will be a big loss for us," Smith added.
"He has played beautifully at the top of the order on some difficult batting wickets. They have been quite slow ... so far throughout the series.
"It's about adapting and getting yourself in. It might take a little bit longer than normal but once you have got yourself in you have to cash in because it's difficult for the new batsman."
Warner is looking at the positives of his enforced absence after a busy stretch of cricket, having led Sunrisers Hyderabad to the Indian Premier League title two weeks ago.
"It's obviously disappointing to be sidelined given the form I've been in," Warner said in a statement.
"(But this) will give me the chance to freshen up and spend quality time with my family after an intense period of action."
Australia next meet West Indies in St Kitts on Monday with the triangular series final on June 26.
Their three-test series against Sri Lanka begins on July 26 and the tour includes five one day internationals and two Twenty20 clashes.
(Reporting by Andrew Both in Cary, North Carolina and Greg Stutchbury in Wellington; Editing by Peter Rutherford)
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