Does the Catholic Church have to submit to an approved private revelation? ...Hence, public revelation is the deposit of faith and rule of faith
  and must be lived by all Catholics.[14] Saint Thomas Aquinas taught
  that all public revelation ended with the death of Saint John the
  Apostle. ... Private revelations cannot surpass, correct, improve, fulfill, complete, or perfect public revelation.


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If the Catholic Church approve a private revelation as worthy is the Church also obliged to follow the messages and requests?

I'm referring to the Fatima visions as they were approved by the Church.
In these visions, the Virgin Mary asks for the consecration of Russia. Is the Church obligated to do it or can the Church (pope) say this is a private revelation and we as the Church don't submit to its authority? Something like if God wants it the Holy Spirit would lead the Pope and the bishops the "standard" way as we know it from the tradition, similarly when they decide dogmas.
No, the Church does not have to submit to private revelation, even if they are approved and are deemed worthy of belief.

Private revelation is outside the deposit of faith and thus the Catholic faithful is not obliged to submit to a particular revelation. Nevertheless some prudence must be applied in whether or not the faithful should believe a particular private revelation.


  67 Throughout the ages, there have been so-called "private" revelations, some of which have been recognized by the authority of the Church. They do not belong, however, to the deposit of faith. It is not their role to improve or complete Christ's definitive Revelation, but to help live more fully by it in a certain period of history. Guided by the Magisterium of the Church, the sensus fidelium knows how to discern and welcome in these revelations whatever constitutes an authentic call of Christ or his saints to the Church. 
  
  Christian faith cannot accept "revelations" that claim to surpass or correct the Revelation of which Christ is the fulfillment, as is the case in certain non-Christian religions and also in certain recent sects which base themselves on such "revelations". - Catechism of the Catholic Church
  
  84 The apostles entrusted the "Sacred deposit" of the faith (the depositum fidei), contained in Sacred Scripture and Tradition, to the whole of the Church. "By adhering to [this heritage] the entire holy people, united to its pastors, remains always faithful to the teaching of the apostles, to the brotherhood, to the breaking of bread and the prayers. So, in maintaining, practicing and professing the faith that has been handed on, there should be a remarkable harmony between the bishops and the faithful." -  Catechism of the Catholic Church


In deciding whether to "submit" to certain demands of a private revelation, the Catholic faithful must exercise the use of prudence.


  Although an assent of Catholic faith may not be given to revelations thus approved, still, an assent of human faith, made according to the rules of prudence, is due them; for according to these rules such revelations are probable and worthy of pious credence. [De Serv. Dei Beatif.] - Apparitions/Private Revelations


Although the Apparitions of Fatima (1917) are in the domain of private revelation and as thus are not part of the deposit of faith, but are worthy of belief, still the virtue of prudence must be exercised by the faithful (pope included) in making an informed decision whether to accept this private revelation or not. 

Since the Visions of Fatima ask that the pope in union with all the bishops of the world to consecrate Russia to Her Immaculate Heart, the question of the prudence in fulfilling this request rests with them.


  Private revelations do not form part of the deposit of faith of the Catholic Church, and its members are not bound to believe in any of them. (Assent may be given based on the discernment of the Church and its judgment that an apparition is probable and worthy of pious credence.) The reported visions at Fátima gathered widespread attention, as numerous pilgrims began to visit the site. After a canonical inquiry, the Bishop of Leiria-Fátima officially declared the visions of Fátima as "worthy of belief" in October 1930, officially permitting the cult of Our Lady of Fátima. - Our Lady of Fátima (Wikipedia)