Catholics haven't chosen anything. It is God the Holy Spirit that guides the papal election the same as it did when Matthias was chosen to succeed Judas Iscariot in Acts 1:26.
For whatever reason lot fell on Francis and he is Pope to all Christians including Protestants whether they recognize it or not. That said, no pope is or will ever be a non-Catholic according to the Sacraments and public profession despite their words and deeds after the fact. One cannot sit on the Chair of Peter and profess something other than full communion with Christ and His Holy Church though they're personal, private actions may dictate otherwise.