Written in 2020; updated.
‘The rosary is wonderful: why can’t we promote it among all Christians?’ The Byzantine Rite used by Orthodox and some Catholic churches has its own devotions to Mary including its own version of the Hail Mary: ‘Rejoice, O virgin Mother of God, Mary full of grace, the Lord is with thee. Blessed art thou amongst women, and blessed is the fruit of thy womb, for thou hast borne the Saviour of our souls.’ There are long litany-like akathists and canons. There is also the prayer rope counting the Jesus Prayer, ‘Lord Jesus Christ, Son of God, have mercy on me, a sinner’, but unlike the Roman rosary it’s not usually a lay devotion at home and in the parishes but a monastic practice. Here is a similar answer from one of the Orthodox churches in America. Like the Stations of the Cross, the rosary developed in the Roman Church after the split with the Orthodox. It began as a substitute for the 150 psalms for the illiterate.
Private prayer can be anything you want. Rite controls what you do in church, to keep good order and to respect various cultures. But a rite is not a costume or just a style of doing church services; it is a package deal, a way of life, a school of theology and spirituality. The Eastern rites, such as the best known among them, the Byzantine Rite, are entirely Catholic as they are. Eastern Catholic churches in their public practice are supposed to model Orthodox and other Eastern Christian life in a reunited church. Members of the Eastern churches do NOT need to adopt Roman Rite devotions in order to be Catholic!
‘But Our Lady says so! She ordered it at Fatima! Russia must be consecrated!’ Approved private revelation such as apparitions is not Catholic doctrine. You don’t have to believe it. Russia was consecrated when St. Vladimir was baptized. The last thing we want to show the Russians is bishops of a foreign church and rite seeming to claim ownership of their country. That wouldn’t reflect our teachings.
The rosary is an okay prayer aid of the Latin Church that works for some people; it is not a sacrament, not a requirement instituted by Christ such as baptism and the Eucharist. It's also a beloved symbol of Roman Catholic identity, often a metaphor for prayer itself: for example, generations of persecuted Irish gathering secretly to pray the rosary. But we're not Roman Catholics.
Delatinization is not anti-Westernism, even though they look alike. I'm not calling Western Catholics heretics.
The most delatinized, liturgically Orthodox priests in the Byzantine Catholic churches were trained in... Rome. That's a clue to the mind and heart of the church on the matter.
Respect for rites goes both ways. If you want to use the rosary in your private prayer, you need not pretend it’s Eastern by byzantinizing it. There is more than one good original Western version of it, such as this and the better known version.
DEUS, cuius unigenitus per vitam, mortem et resurrectionem suam nobis salutis aeternae praemia comparavit, concede, quaesumus: ut haec mysteria sacratissimo beatae Mariae Virginis rosario recolentes, et imitemur quod continent, et quod promittunt assequamur. Per eundem Christum Dominum nostrum. Amen.
No matter how you prefer to pray privately, God loves you and may he bless you!