Can only the priest lead the ritual of washing the feet in Holy Tursday in the Catholic Church?
The short answer is yes, only a bishop or priest may preform the Mandatum.
According to the rubrics only bishops and priests are liturgically permitted to wash the feet of the faithful. It should be noted that the rite of Mandatum remains optional during the Holy Thursday celebration of the Mass.
A decree published on January 6, 2016 modified the liturgical Mandatum:
In the Roman Liturgy this rite was handed down with the name of the Mandatum of the Lord concerning fraternal charity from the words of Jesus (cf Jhn 13:34), which are sung in an Antiphon during the celebration.
In performing this rite Bishops and priests are invited to intimately conform themselves to Christ who «came not to be served but to serve» (Mt 20:28) and, compelled by charity «to the end» (Jhn 13:1), to give his life for the salvation of the whole human race.
In order that the full meaning of this rite might be expressed to those who participate it seemed good to the Supreme Pontiff Pope Francis to vary the norm which is found in the rubrics of the Missale Romanum (p. 300 n. 11): «The men who have been chosen are led by the ministers…», which therefore must be changed as follows: «Those who are chosen from amongst the people of God are led by the ministers…» (and consequently in the Caeremoniali Episcoporum n. 301 and n. 299b: «seats for those chosen»), so that pastors may select a small group of the faithful to represent the variety and the unity of each part of the people of God. Such small groups can be made up of men and women, and it is appropriate that they consist of people young and old, healthy and sick, clerics, consecrated men and women and laity.
This Congregation for Divine Worship and the Discipline of the Sacraments, in virtue of the faculties granted by the Supreme Pontiff, introduces this innovation into the liturgical books of the Roman Rite, reminding pastors of their responsibility to adequately instruct both the chosen faithful as well as all others so that they may participate consciously, actively and fruitfully in the rite. - Decree: In Missa in Cena Domini
Thus it is clear that only bishops and priests may perform this liturgical action. However those whose feet are to be washed is no longer reserved to men. Women may now be included. The number of persons has been set up to a maximum of 12.
Nota Bene: Girls are permitted to serve as altar servers in Ordinary Form of the Mass.