Some people say that Mary promised St. Dominic, "Whatever you ask in the Rosary will be granted." According to the Church, is this statement an authentic quote, and if so, what is the teaching behind it?
Why would the Catholic Church make an explication on such a statement when such a tradition is in serious doubt?
The Catholic Encyclopedia has a more reserved approach to its' origin as a whole and tends to see the rosary as a natural development of piety of which St. Dominic does not have any particular role at all:
We have positive evidence that both the invention of the beads as a counting apparatus and also the practice of repeating a hundred and fifty Aves cannot be due to St. Dominic, because they are both notably older than his time. Further, we are assured that the meditating upon the mysteries was not introduced until two hundred years after his death. What then, we are compelled to ask, is there left of which St. Dominic may be called the author?
These positive reasons for distrusting the current tradition might in a measure be ignored as archaeological refinements, if there were any satisfactory evidence to show that St. Dominic had identified himself with the pre-existing Rosary and become its apostle. But here we are met with absolute silence. Of the eight or nine early Lives of the saint, not one makes the faintest allusion to the Rosary. The witnesses who gave evidence in the cause of his canonization are equally reticent. In the great collection of documents accumulated by Fathers Balme and Lelaidier, O.P., in their "Cartulaire de St. Dominique" the question is studiously ignored. The early constitutions of the different provinces of the order have been examined, and many of them printed, but no one has found any reference to this devotion.
Wikipedia goes on to say:
Alanus de Rupe (Alain de la Roche) was a 15th-century Dominican preacher, best known for his efforts to promote the Rosary. Alanus claimed to have experienced a vision by which it was revealed to him that the Blessed Virgin Mary had appeared to St. Dominic and gave him the Rosary as a means to combat the Albigensian movement. Alanus de Rupe's revelation concerning St. Dominic and the Rosary was generally accepted until the 17th century when the Bollandists concluded that the account of Dominic's supposed apparition of Our Lady of the Rosary is not mentioned in any documents of the Church or Dominican Order prior to the accounts of Alanus over two hundred years later.