Raonic, seeded 11th, won his match after it was interrupted by rain in the second set.
He defeated Victor Hanescu of Romania 7-6 (9), 6-4, 6-4 and will next face Grigor Dimitrov of Bulgaria, who defeated Lu Yen-Hsun of Taiwan, 6-3, 6-3, 7-6 (11).
Pospisil, the No. 28 seed, pulled out during a doubles match a day before his scheduled third-round singles match against Stanislas Wawrinka of Switzerland.
Pospisil, 23, and Austrian partner Julien Knowle were losing 1-2 to an Australian duo in the first round of their match when he withdrew after being told by doctors he risked compromising his season if he continued.
He will return to Vancouver on Friday to start rehab in hopes of playing in the Davis Cup in Tokyo in two weeks.
Pospisil called it an unfortunate and disappointing injury.
"I'm really disappointed to be pulling out. It's my first third round in a Grand Slam. I was excited to do my best and take the opportunity, but unfortunately not able to do so at this time."
"It's nothing too serious right now, but there's significant risk if I were to play on it that I could do some serious damage, Pospisil added.
"I don't want to jeopardize the rest of my season in any way. So it was just the smart move really to pull out, and be safe with my body and try and get ready hopefully in time for Davis Cup."
Pospisil first felt the problem in a semi-final against Wawrinka this month in Chennai, India and withdrew in the second set. He also withdrew from a tournament last week in Sydney.
Pospisil bounced back on Wednesday, however, to earn a 3-6, 7-6 (6), 7-6 (9), 6-1 victory over Australian Matthew Ebden.
"There's nothing that's really wrong with it (the back) right now," stressed Pospisil. "It's just more preventive, I'm supposed to keep off it for a little while."
Raonic of Thornhill, Ont., began his match in the evening after tournament officials had halted play on all courts except the covered ones due to a temperature of more than 43 C on the third day of a record heat wave.
Raonic, 23, is into the Melbourne third round for a fourth consecutive year.
"It was a day with a lot of different adversities, it wasn't too difficult of a day,"said the Canadian. "I'm just happy I got through it as efficiently as I did."
"Before the rain the conditions were quite different, a lot quicker. Then I come out starting from the first game getting everything figured out. The lighting is a little bit yellow. Pretty much high tosses get lost a little bit because the lights only face down. There's no upwards or sideways lighting."
No. 76 Hanescu had defeated Raonic in the first round of the Rogers Cup in Toronto in 2010 but lost their next meeting in Chennai at the start of 2012.
Raonic found himself in a fight early on with Hanescu as the pair each held onto serve in a set which eventually went into a tiebreak.
The Canadian saved a Romanian set point but needed three of his own to earn the set after 56 minutes.
In the second set, Raonic went up an early break and was in control when the weather moved in. He returned to take the set, stayed in command in the third and posted the victory in two hours, 19 minutes with 27 aces on his fourth match point.
Raonic is keen to take on Dimitrov: "A difficult match, very talented, very capable. He can do a lot of different things. I have to try to take control as much as I can and try to dictate as much as possible."