Raonic, 26, injured an abductor muscle in his Australian Open loss to Rafael Nadal and has pulled out of the Davis Cup tie from 3-5 February in Ottawa.
Britain are without world number one Andy Murray after he was not included in Leon Smith's squad, giving him time to recover after the Australian Open.
"I am sad and disappointed that I am unable to compete," said Raonic.
Raonic's absence means Canada do not have a top-100 singles player, with world number 132 Peter Polansky joining 135th-ranked Vasek Pospisil and Wimbledon junior champion Denis Shapovalov in the team.
Veteran doubles player Daniel Nestor completes the Canada team, which is captained by Martin Laurendeau.
Kyle Edmund, the world number 46, and 51st-ranked Dan Evans will play singles for Great Britain, with Jamie Murray and Dom Inglot teaming up for the doubles.
The winners will face France in the quarter-finals in April, with Britain facing an away tie should they progress.
World number three Milos Raonic will miss Canada's Davis Cup tie against Great Britain next week through injury.