Well, I think the two are on separate planes.
The Eastern Churches expect all Christians to invoke constantly the name of Jesus in the Jesus Prayer (as did St John Chrysostom and the other Fathers).
But let’s take a look at the Rosary itself. It originated in the Coptic Thebaid where saints would use various types of prayer counters (including knotches on their staffs) to pray 150 Our Fathers and Hail Marys and other prayers as a kind of replacement to the Psalter itself (although there were also those who memorized one psalm and recited it 150 times).
St Seraphim of Sarov was actually not, in all likelihood, influenced by the Western Rosary. The reason is that he not only promoted the 150 Hail Marys divided into decades, but also the ancient 150 Our Father’s or “Pater Noster Psalter” as it was called in the West. This is why in St Seraphim of Sarov’s version of the rule of the Theotokos, there is only one Our Father at the beginning, but none at the head of each of the decades . . .
And St Seraphim of Sarov did NOT include mysteries to meditate on during the recitation of his rule of the Theotokos - that was done by St Seraphim Zvezdinsky, a Russian Old Believer Archbishop (who was in union with the ROC), who was shot by the Bolsheviks in 1937 and who had a great veneration for St Seraphim of Sarov (whose name he took as a bishop and whose icon of the Virgin of Tenderness he wore as his episcopal Panaghia).
This New Martyr liked to include a Troparion that most closely coincided with the Mystery under consideration and this is the version that is included in the prayerbook I have but also in the Encyclopedia of Orthodoxy published at Moscow in 2003.
St Seraphim of Sarov, great mystic he was, said that he received visions from our Lady who urged him to promote the Rule/Rosary of the Theotokos and that THIS form of prayer is one that she places higher than any other as a way to invoke her protection over our lives and her intercession on our behalf!
At Diveyevo Monastery, there is the tradition of walking around the pathway that surrounds the great Monastery while reciting the Rule of the Theotokos (in some cases, I’ve read that this also includes the 150 Our Fathers). The nuns there do it together and on feast days they sing the Hail Mary’s.
But this is a private, not a liturgical prayer practice and is never recited in Church as such. It would not be in keeping with Eastern Church practice to pray it out loud in Church before Divine Liturgy for instance - no one is stopping anyone from praying it in private or in a group outside of that context.
There is also nothing wrong about adopting the Western Mysteries of the Rosary including the Mysteries of Light.
I love reciting all 20 decades of the Rosary daily with additional “tags” reflecting the Mysteries inserted toward the end of our Hail Mary. So far, I’ve felt no inclination to cross myself in the Latin way as a result, or to genuflect or even to take up the study of Latin so I can pray the Tridentine office . . .
There were Orthodox Saints who also prayed the Hail Mary as their form of the Jesus Prayer i.e. they continually recited the Hail Mary and were called “Elders of the Theotokos.” In the Way of the Pilgrim, the pilgrim meets a fellow who said the Hail Mary daily and he advised him to “always and continually pray that prayer to the Most Holy Theotokos.”
We shouldn’t worry - the Mother of God always takes us to her Son and when we are filled with thoughts of her and how glorious she is, that is when our Lord particularly finds our souls a worthy abode for Him! As St Theophan the Recluse said, just as mothers nourish their earthly children with their milk from their bodies, so too does the Mother of God nourish us, her children, through Holy Communion . . . Devotion to her, he wrote, is not like devotion to any other Saint . . .
As Phillip the Master Beadsman said, RC devotion to the Most Holy Name of Jesus is very inspiring. St Francis of Assisi, for instance, whenever he said the Name of Jesus, would stop and then put out his tongue to lick his lips. He said that when we say the Name of Jesus reverentially, it is like spiritual honey exploding in our mouths and we must take care to lick up any that might have dripped out onto our lips . . .
Alex
I remember hearing somewhere that Mother Teresa claimed to be always saying the Rosary. She prayed it so much that it became grafted to her heart (to use Eastern terminology). But you are certainly correct. The rosary and the Jesus Prayer are just different. One is not worse than the other. I believe even some of the Orthodox Fathers would be appalled at the thought of folks claiming one is superior to the other. For them prayer is prayer and all prayer has but one goal; communion with God.
I didn’t mean to at all. I was referring to a general sense that I’ve come across from others who promote the Jesus Prayer in contrast to the Rosary. But, as you and I both know, the there is no contrast or contradiction between the two, only the full harmony of a well-tuned symphony. 
And “Warm Alex” was my late father’s nicname for me - it is actually the popular name for today’s old calendar feast of “Warm Alex” (my Namesday) since st Alexius the Man of God’s day was usually a warm one (and today it is not).