Carlsey's caretaker spell ends with today's Nations League clash with the Republic of Ireland.
Redknapp wrote for The Sun: "If we’ve learnt anything from Lee Carsley’s time as interim boss, it is this — he’s not cut out to be an England manager.
"A great coach, yes — his record with the Under-21s is proof of that. Not scared of making bold decisions, either, as he showed by leaving Harry Kane on the bench in Greece on Thursday.
"I wasn’t the only one surprised to see Ollie Watkins in the side instead. But after a win like that, you can’t argue with him.
"It was a fantastic result, for sure. But, with all due respect, it doesn’t mean he should have got the job on a full-time basis. There’s no doubt he could have done at one stage.
"However much the FA insist Thomas Tuchel was always their first choice to replace Gareth Southgate, it’s obvious he wasn’t.
"It was Lee’s to lose and he did exactly that with the first result against Greece — and his interview afterwards did even more damage."