No!
Newt had to obtain annulments of his prior marriage(s) in order to marry Callista in the Catholic Church.
But if Newt had never become a Catholic, he and Callista would still need an annulment before they could marry in the Catholic Church.
There two are exceptions to this: If a person becomes a Catholic and the existing spouse refuses to live in peace with the convert. I believe this is called the “Pauline privilege.” There’s something called the “Petrine privilege,” but I forget what it is.
Sounds like you know more than I do.