Bhambri was forced to retire early due to injury, just 45 minutes into the popular men's tune-up event for Wimbledon, which starts July 2, reported Efe.
Before Bhambri's withdrawal, Raonic was already on his way to a win, having fired 13 aces compared to zero from his 25-year-old Indian opponent.
Raonic, 27, is next set to take on the winner of the clash between David Goffin of Belgium, world No 9, and Feliciano Lopez of Spain, world No 37.
Meanwhile, former world No 1 Novak Djokovic of Serbia defeated John Millman of Australia 6-2, 6-1 to advance to the second round, marking a winning return to the tennis tournament.
The Serbian star, currently world No 22, needed a brisk 64 minutes to beat his world No 63 opponent.
Djokovic, 31, is competing in the tournament for the first time since 2010, and was runner-up to Spanish tennis great Rafael Nadal in the 2008 final.
The 12-time Grand Slam champion is set to play the second round against 2014 Queen's Club champion Grigor Dimitrov of Bulgaria, world No. 3, who defeated Damir Dzumhur of Bosnia and Herzegovina 6-3, 6-7(4-7), 6-3.
Dimitrov said: "It was one of those matches that the only thing I am happy with is just winning.
"I definitely could have done a few things better, but it is still my first match on grass. I was a little rusty, obviously. Overall I served well and returned well when I had to. Those are all good signs for me. I just want to pick up that positivity and carry it on," he said.
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