Carter-Vickers complained about having pain in the match, Rodgers said, but the manager ran out of substitutions and elected to keep the defender on the pitch while Celtic was clinging to a one-goal lead.
"We wanted to take off Cameron, but we could only make one substitution in the second half. Ideally, we would have taken him off," Rodgers said after the match.
On Friday, he told Sky Sports he expects Carter-Vickers to miss significant time.
"He looks like he's done his Achilles, which could be anywhere between three and five months," he said. "We just await confirmation of that, but it's not great news."