Finch's injury will be assessed on a "day by day" basis but with the three matches being played in the space of five days it could be a challenge for him to fit for this series.
Carey was previously a joint vice-captain of the ODI side until the selectors reverted to the traditional model of having one player in the position by naming Pat Cummins as vice-captain across all formats last year.
However, Cummins is not part of this tour so Carey now has the chance to lead his country for the first time having previously captained Australia A, Adelaide Strikers and South Australia.
"I am deeply honoured to be able to lead the team while Aaron recovers," he said. "To captain Australia is one of the greatest privileges in sport and an honour I am extremely thankful to receive.
"Finchy is our captain and we will welcome him back with open arms when he is fully fit, so for now I hope I can fill the role to his extremely high standards.
"To lead this group against the West Indies in the West Indies is a great challenge and one I very much look forward to for as long as I have the opportunity."
Carey has previously been suggested as a possible candidate to captain the Test side when Tim Paine retires and his leadership qualities are held in high regard. However, last season he drew strong criticism from Allan Border for the way he led Australia A on the second day of their match against India at the SCG although afterwards the pair had a conversation.
"My takeaway is that there's a lot of really great leaders throughout the country still caring about the game of cricket," Carey said on Monday. "For him to answer my call and have a chat about what he saw, that was a great learning curve for me to speak to one of Australia's best captains. I didn't take it too much to heart and saw it as more of an opportunity to learn."
Justin Langer, the Australia head coach, said: "For a long period of time Alex has shown an extremely high level of professionalism and discipline as a fantastic leader within this group. The injury to Finchy gives Alex an opportunity for his first experience as captain. I have no doubt he will do a fantastic job with the support of the other senior players."
The absence of Finch will leave Australia with an even more inexperienced side having been stripped of a host of key names for this tour. Carey did not confirm an XI for the opening match but indicated he would be batting in the middle order.