The former Chelsea and Bayern Munich head coach will be the new Three Lions manager until the end of the 2026 World Cup.
Tuchel's appointment as England manager has proved divisive, with many feeling an English manager should have been chosen.
Carragher told Sky Sports: “My feeling is, the whole point of international football, certainly with major nations who compete for the tournaments, is that it's your best versus their best.
“Club football is different and we love it, different nationalities, different coaches in the Premier League or at Champions League level. It's a real mix.
“But the thing that makes international football different from the club game is that it's people from your country.
“I know that's skewed a little bit with players, maybe they weren't born in the country or it goes back to maybe their grandparents.
“And I also understand, certain nations having a foreign coach who's going to have an influence them will drag the whole footballing structure up with them and pass on experience and expertise to all the coaches throughout the system and the country.
“But when I think of England now, and we're so close to winning a major tournament, there's so much good work that has gone into producing these players at St George's Park, it just doesn't feel right to me that we have a foreign coach.
“Again, it's not just about England. I don't think Italy should, I don't think Germany should, I don't think France should. Portugal have it at the moment in Roberto Martinez which I find strange.
“That's my take on it. I'm not going to lose any sleep over it. England have got a brilliant manager, but I think England should have an English manager.”