The first definite evidence for the promotion of what corresponds to the modern Rosary is found in the second half of the fourteenth century, in the work of Alan de la Roche and his fellow Dominicans, but traditionally this devotion goes back to the time of St Dominic himself or even earlier.
According to Alan, Dominic had revived the practice of saying the Rosary in response to revelations from the Blessed Virgin while he was engaged in his fight against the Albigensians, as a means of winning them back to the Church.
The Albigensians opposed Church authority, holding a dualistic view of reality with two 'gods,' one in which the spiritual realm had been created by the good deity and matter by the bad. They rejected the Sacraments as well as many basic Christian principles, including the resurrection of the body, and adopted an extremely rigorous view of life which condemned marriage, while they also favoured a form of suicide by starvation.
These ideas were considered a clear threat not only by the Church but also by society as a whole, and were condemned by numerous Church Councils. Despite this the movement grew rapidly and missionaries including Dominic were sent to convert them. He had partial success, but a Crusade, in which Dominic took no part, had to be launched against them, and this coupled with the Dominican Inquisition managed to destroy this heresy by the end of the fourteenth century.
Dominic's part in the development of the Rosary has been disputed, but there is no question that there has been a long-standing tradition in the Church which regards this particular form of meditative prayer as the best form of devotion to Mary, and hence ultimately to God, since prayer to Mary is not an end in itself, but leads to Christ.
The rosary has been criticised on the grounds that there is insufficient documentary evidence as to its beginnings, but given the extraordinary degree of later Church approval at the highest level, particularly from the Popes, this criticism is not justified. With only a couple of exceptions all the Popes from the late fifteenth century until now have acclaimed the rosary with its mixture of vocal and mental prayer.
It seems that there is a definite tradition within the Dominican order linking the founder to the propagation of the rosary, and that possibly Dominic may have preached sermons on the basics of the faith and interspersed these with 'Hail Marys', thus initiating the idea of 'meditating' on the mysteries surrounding the life, death and Resurrection of Jesus.
The complete Rosary consists in the recitation of fifteen groups of ten 'decades' of the Marian prayer the 'Hail Mary', each headed by an 'Our Father', while meditating on the principle events surrounding Jesus' life, especially where these have a specific connection with Mary.
The Rosary then consists of 150 Hail Mary's and this indicates its origin probably lay as a counterpart to the 150 psalms which were recited by religious orders as part of the Divine Office. Generally it is said in three groups of fifty Hail Mary's to correspond to the 'Joyful', 'Sorrowful' and 'Glorious' mysteries of Christ's life, death and Resurrection, which were lived in union with Mary his Mother.
In essence the Rosary is a prayerful Scriptural meditation, since the Our Father is Jesus' own prayer given to his disciples when they had asked him how they should pray (Matt 6:9-13). The first part of the Hail Mary is also Scriptural, being a compilation of part of the dialogue between Mary and the Angel Gabriel at the Annunciation (Lk 1:28), combined with the exclamation made by Elizabeth during the Visitation (Lk 1:42). In fact the prayer was originally known as the 'Angelic salutation' (greeting), with Elizabeth's greeting only being added generally during the medieval period.
The second part of the Hail Mary, the intercessory prayer to Mary, seems to date from about the eleventh century and was gradually adopted by the Church in general, with the whole prayer being finally fixed in its present form during the sixteenth century.
Sources: Carol, Mariology, Vol. 3; Cross, Oxford Dictionary of the Christian Church.
Homiletic & Pastoral Review article:
St Dominic and the Rosary, by Paul A. Duffner
It is very probable that the Mother of God did use St. Dominic in some way to give the Rosary to the Church.