Is the Eucharist more holy than the other Sacraments in the Catholic Church?

Not all sacraments are equally holy:

Council of Trent, session 7, canon 3
If anyone says that these seven sacraments are so equal to each other that one is not for any reason more excellent [or "more worthy", dignius] than the other, let him be anathema.

St. Thomas Aquinas ordered the sacraments this way (Summa Theologica III q. 65 a. 1 c.):

We may likewise gather the number of the sacraments from their being instituted as a remedy against the defect caused by sin.

1. Baptism is intended as a remedy against the absence of spiritual life;
2. Confirmation, against the infirmity of soul found in those of recent birth;
3. Eucharist, against the soul's proneness to sin;
4. Penance, against actual sin committed after baptism;
5. Extreme Unction, against the remainders of sins—of those sins, namely, which are not sufficiently removed by Penance, whether through negligence or through ignorance;
6. [Holy] order[s], against divisions in the community;
7. Matrimony, as a remedy against concupiscence in the individual, and against the decrease in numbers that results from death.

For the ordering above, he gives this reason (ibid. a. 2 c.): "those sacraments which are intended for the perfection of the individual naturally precede those which are intended for the perfection of the multitude". Also, Holy Orders and Matrimony are last because they are "intended for the perfection of the multitude", and "Matrimony is placed after order, because it has less participation in the nature of the spiritual life, to which the sacraments are ordained." He says Eucharist could be placed before Confirmation, too, because nourishment causes growth. Baptism is so important! Everything is rooted in it!

He gives an alternative ranking, pairing the sacraments with specific virtues (ibid. a. 1 c.):

Some, again, gather the number of sacraments from a certain adaptation to the virtues and to the defects and penal effects resulting from sin. They say that

1. Baptism corresponds to Faith, and is ordained as a remedy against original sin;
2. Extreme Unction, to Hope, being ordained against venial sin;
3. Eucharist, to Charity, being ordained against the penal effect which is malice.
4. [Holy] order[s], to Prudence, being ordained against ignorance;
5. Penance to Justice, being ordained against mortal sin;
6. Matrimony, to Temperance, being ordained against concupiscence;
7. Confirmation, to Fortitude, being ordained against infirmity.

Is the Eucharist more holy than the other Sacraments in the Catholic Church?

The short answer is yes.

Pope St. Paul VI stated that the Eucharist is the apex of our faith and Pope Benedict XVI stated that the Eucharist is the apex of our prayer!

The Holy Eucharist, Vatican II tells us, is "the source and summit of the Christian life", (Lumen Gentium no. 11; cf. no. 1324). Since the Christian life is essentially a life, we might say as well that the Eucharist is the "source and summit of Christian spirituality" too.

Taking part in the Eucharistic sacrifice, which is the fount and apex of the whole Christian life, they offer the Divine Victim to God, and offer themselves along with It. Thus both by reason of the offering and through Holy Communion all take part in this liturgical service, not indeed, all in the same way but each in that way which is proper to himself. Strengthened in Holy Communion by the Body of Christ, they then manifest in a concrete way that unity of the people of God which is suitably signified and wondrously brought about by this most august sacrament. - Lumen Gentium promulgated by Pope Paul VI.

Of all the sacraments, only the Eucharist is called the Most Holy Sacrament*.

The Holy Eucharist is the greatest of all the sacraments.

Baptism is considered the most necessary sacrament for without Baptism we cannot get to heaven. Yet, despite all the wonderful things that Baptism and the other sacraments accomplish in the soul, they still are but instruments of God for the giving of grace.

But the Holy Eucharist is not merely an instrument for the giving of grace—here is the actual Giver of grace Himself, Jesus Christ our Lord truly and personally present.

The Sacrament of the Holy Eucharist is central to our Catholic faith. Belief in the Eucharist as the True Body and Blood of Christ is essential to being a Catholic.

This belief was evident in the writings of St Paul and the patriarchs of the early Church and went largely uncontested until the Reformation in the 16th century. Even then, Martin Luther who is considered the father of the Protestant reformation believed in the true presence of Christ in the Eucharist.

Our belief in the Holy Eucharist is an act of faith. The Eucharist is a mystery that cannot be proven scientifically. It is beyond human understanding. St Thomas Aquinas attempted to explain the process using the term transubstantiation. The substance of the bread and wine, by the power of the Holy Spirit, becomes the true substance of Christ without a change in the outward appearance or physical properties of bread and wine.

Though it is a matter of faith, our belief of the True Presence in the Eucharist has strong scriptural support.

In the Bread of Life discourse, Gospel of John chapter 6, Jesus tells his disciples:

"I am the bread of life. Your ancestors ate the manna in the desert, but they died; this is the bread that comes down from heaven so that one may eat it and not die. I am the living bread that came down from heaven; whoever eats this bread will live forever; and the bread that I will give is my flesh for the life of the world.” The Jews quarreled among themselves, saying, “How can this man give us [his] flesh to eat?Jesus said to them, “Amen, amen, I say to you, unless you eat the flesh of the Son of Man and drink his blood, you do not have life within you. Whoever eats* my flesh and drinks my blood has eternal life, and I will raise him on the last day. For my flesh is true food, and my blood is true drink. Whoever eats my flesh and drinks my blood remains in me and I in him. Just as the living Father sent me and I have life because of the Father, so also the one who feeds on me will have life because of me. This is the bread that came down from heaven. Unlike your ancestors who ate and still died, whoever eats this bread will live forever.”

Many followers rejected Jesus that day for they felt that his statements were too hard to accept. Some thought he was a madman. However, Jesus did not recant or state that he was speaking symbolically. He stood his ground and let them walk away. He was speaking the literal truth.

In the synoptic gospels of Matthew, Mark, and Luke we read Jesus' words from the Last Supper, words that we hear proclaimed at Mass as part of our Eucharistic prayer:

"While they were eating, Jesus took bread, said the blessing, broke it, and giving it to his disciples said, “Take and eat; this is my body.” Then he took a cup, gave thanks, and gave it to them, saying, “Drink from it, all of you, for this is my blood of the covenant, which will be shed on behalf of many for the forgiveness of sins."

It was at the Last Supper, on the night he was betrayed, that our Savior instituted the Eucharistic sacrifice of his Body and Blood. This he did in order to perpetuate the sacrifice of the cross throughout the ages until he should come again, and so to entrust to his beloved Spouse, the Church, a memorial of his death and resurrection... CCC 1323