Rangers could be crowned Scottish champions today under Gerrard, who continues to be linked with a return to Anfield.
But Carragher told the Sunday World: "If Gerrard continues to win big in Scotland, it is sure to land him many Premier job offers, with the implication that managing a mid-table team in England is more attractive than being Rangers or Celtic coach.
"I am not convinced that applies in Gerrard's case. There was a time when I might have gone along with the idea that Rangers is a good first stepping stone for my former Liverpool team-mate, and he should have his eye on a swift Premier League return in readiness for an inevitable Anfield reunion.
"Having seen what he is building at Rangers, I do not believe he has to leave Ibrox and prove himself anywhere else. Certainly not yet.
"He has fallen in love with the club and the city of Glasgow and looks to be settling in for the long haul.
"Anyone who knows Stevie well need only look at his reaction against Livingston at half-time on Wednesday, when he was sent off for being livid at the decision not to award Alfredo Morelos a penalty.
"That's how he is when dedicated to a cause. Fully committed. Despite the title being done and dusted, every point matters. What's so impressive about Gerrard's success is the manner in which he has achieved it. When he made the move, it took guts. Whoever took on the Rangers job two-and-a-half years ago had to focus on an immediate objective of preventing Celtic winning 10 titles in a row. Gerrard was going to be judged on that, not as a novice coach making his way."