No. You really need the number after the name,
St John Paul II. The number does not get attached because he is
the second Saint John Paul, but because he is
the second pope named John Paul (and has been canonized a saint).
In the calendar, he will be listed in the style of other sainted popes:
Saint John Paul II, pope. In everyday speech (if you are the habit of speaking of sainted popes everyday) it is common to style such saints as
Pope Saint John Paul II (but it is not incorrect to do otherwise, unless omitting the number*).
People of such a mind may substitute
the Great for the number
II – Such title is a popular acclamation without any official procedure. None of the other “Great” sainted popes is known by that appellation in any such capacity.
(* Or the title “the Great”

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