The tourists recorded two positive tests while five members of their squad are already in isolation and under the supervision of the medical team.
Ireland were hoping for a busy schedule in the West Indies after their one-day international series against the United States was cut short last month due to Covid-19.
The team were scheduled to play three ODIs - they lost the first by 24 runs - and one T20 match.
A joint statement from Cricket Ireland and Cricket West Indies said: "Cricket West Indies (CWI) and Cricket Ireland (CI) issued a joint statement after two further positive Covid-19 cases were reported in the Ireland camp following recent testing, and two players being ruled out due to injury, resulting in Ireland having a severely depleted squad.
"The second CG Insurance one-day international scheduled for Sabina Park on Tuesday, January 11 has therefore been postponed.
"CWI and CI are in discussions to review the match schedule and are working to explore all possibilities for the series to be completed.
"The five members of the Ireland team who have returned positive Covid-19 results remain in isolation under the care and supervision of the medical teams. Further updates will be provided as soon as possible."