Holness also noted there are "unresolved" issues as he greeted the Duke and Duchess of Cambridge in front of a media scrum.
"We are moving on," he said. "We intend to ... fulfil our true ambitions and destiny as an independent, developed, prosperous country."
The former British colony would become only the second Caribbean island to sever relations with Queen Elizabeth II in recent years, with Barbados doing so in November.
The announcement surprised many on the island of nearly three million people.
"I did not know that the prime minister was going to say what he said today. It is a very important step forward," said Carla Gullota, director of Stand Up for Jamaica, a nonprofit human rights organisation that joined dozens of other groups and leaders in signing a recently published letter demanding an apology and reparations from Britain.
The announcement, which comes a day after Gullota and others joined a protest held hours before the royal couple arrived in Jamaica on Tuesday as part of a weeklong tour of Central America and the Caribbean that coincides with the 70th anniversary of the Queen's coronation.
"This visit has brought back to light that many Jamaicans are looking forward to Jamaica becoming a fully independent republic," she told The Associated Press.
Gullota said she does not support reparations in the form of a cheque, which she called "ridiculous".
Instead, she said Jamaicans should be compensated in other ways with things including scholarships and access to health care.
"What was not offered in the past, should be offered now," she said.
Britain ruled Jamaica for more than 300 years, forcing hundreds of thousands of African slaves to toil the land under brutal conditions.
Jamaica gained its independence in August 1962 but remained within the British commonwealth.
