Neither does JP II, in the sense which you are referring to. Both Leo XIII and JP II hold that "ut unum sint" was fulfilled in the unity of the Catholic Church. That is why, in UUS 9, he points out that to desire unity means to desire the Church. JP II and Leo XIII also hold that "ut unum sint" includes the men now living who will convert to the Catholic Church before their death. That is why, in a sense, the prayer lacks fulfillment (but is fulfilled through ecumenism which returns non-Catholics to the unity of the one Church of Christ which subsists in the Catholic Church), not however in the sense condemned by Pius XI. Otherwise, how could his most august successor state clearly:
40. Truly we are aware of the accumulation of prejudice that tenaciously prevents the happy fulfillment of the prayer offered by Christ at the last Supper to his Eternal Father for the followers of the Gospel: 'That they may be one' (John xvii, 21). But we know also that such is the strength of prayer, when those who pray are joined together in a common fervor, a strong faith, and a clear conscience, that it can lift up a mountain and cast it headlong into the sea (cf. Mark ii, 23). We desire then and we wish that all those who have at heart an earnest invitation to Christian unity -- and surely no one who belongs to Christ would belittle the importance of this matter -- should pour forth their united prayers and supplications to God, from whom comes all unity, order and beauty, that the praiseworthy desires of every right-thinking person may soon be brought to fulfillment. Let research be made without jealousy or anger to straighten out the path by which this good may be reached; let us bear in mind that today we are accustomed to retrace and weigh the events of bygone ages more calmly than in the past. (Pius XII, Sempiternus Rex Christus)
"Let those then who, through the calamities of time, have been cut off, not be slow to pay due respect to this divinely erected and unbroken rock, this Apostolic See for whom to rule is to serve. Let them bear in mind and imitate Flavian, that second John Chrysostom, in his sufferings for justice; and the fathers of Chalcedon, those most worthy members of the Mystical Body of Christ; and Marcian, that strong, gentle and wise ruler; and Pulcheria, that resplendent lily of inviolate royal beauty. From such a return to the unity of the Church we foresee that there would flow a rich fountain of blessings unto the common good of the whole Christian world."
40. Truly we are aware of the accumulation of prejudice that tenaciously prevents the happy fulfillment of the prayer offered by Christ at the last Supper to his Eternal Father for the followers of the Gospel: 'That they may be one' John xvii, 21).
We desire then and we wish that all those who have at heart an earnest invitation to Christian unity —and surely no one who belongs to Christ would belittle the importance of this matter—should pour forth their united prayers and supplications to God, from whom comes all unity, order and beauty, that the praiseworthy desires of every right-thinking person may soon be brought to fulfillment. Let research be made without jealousy or anger to straighten out the path by which this good may be reached; let us bear in mind that today we are accustomed to retrace and weigh the events of bygone ages more calmly than in the past.
And Pius XII goes on and on about the need for all of the separated to return. It's the Catholic ecumenism of return. The only ecumenism that is viable. JPII is not willing to go this far in UUS. He dances, skirts and wastes a lot of peoples time and energy with his indirection and misdirection.