If faith is a gift from God, and atheists (might) go to hell, is God ultimately willing to send atheists to hell? (Catholic perspective)

I found some insightful notes regarding this topic in an article on Grace. It is written by Fr. John Hardon, a Jesuit theologian, Although I cannot tell whether all that he wrote (and quoted below) is official Catholic doctrine, he worked with a few popes, wrote some famous catechisms editions and always remaining in the Orthodoxy. I quote (emphasis in original; bold added):

Infidels make up the majority of mankind. All can be saved; to be saved they must die in sanctifying grace. Therefore all can get sanctifying grace. But how? They must do something, since they are adults: they must dispose themselves for justification, for sanctifying grace. How? By salutary (grace-elevated) acts of faith and fear and repentance and love etc. God, then, must first “lean down” and give them grace, for nature alone cannot produce salutary acts.

Just when does God give the first actual grace to an infidel – which can gradually or quickly lead him to the “big” graces of revelation and faith (i.e. assent to this revelation in a salutary act of faith)? At the time that God judges to be opportune, which according to many theologians is the infidel’s first full use of reason, when he distinguishes between good and evil.

Just what is the first grace God gives to a particular infidel, we do not know. It could take many forms. It might be a grace to turn to God, to acknowledge Him, to express his need of God or of divine help. It could involve both an external grace and an internal grace (e.g. of prayer). But sometime, somehow every adult infidel will get this remote vocation to faith and sanctifying grace. If he cooperates properly with this, God gives him further graces and ultimately the grace that is proximately sufficient for the act of faith, i.e. the grace of revelation and the grace of faith to assent to this revelation.

The article continues on more details about the above, and on salvation outside the Church. Then, summarises:

In Summary, then, we may say that God gives all adult infidels sufficient grace for salvation, and if they use it properly they will be saved.

So, God gives every (adult) human being the first impulse of grace to reach Him. It's our cooperation with Him which will ultimately lead to our justification, and salvation. I'm still pondering on the details of this, but it is in my opinion sufficient structure for an answer; hence this post.

Good question with concise phrasing! To paraphrase, "If faith is God's gift, then those who are not given this gift are not at fault and should not be punished (with Hell)."

Overview

The core of your question deals with the way that man's cooperation is involved in his salvation. It is an age-old question that probably reached it's peak in the Congregatio de Auxiliis dispute between the Dominicans and the Jesuits which took place in the 16th century. It is secondarily addressed in the debates on predestination and free will that have taken place between various denominations and Churches (e.g. Calvin and Pighius).

Thomas Aquinas on Divine Concurrentism

Since you quoted Thomas Aquinas let's make use of his writings to try to get at the Catholic Church's high-level theological answer.

From the part you already referenced:

As regards the second, viz. man's assent to the things which are of faith, we may observe a twofold cause... ...Therefore faith, as regards the assent which is the chief act of faith, is from God moving man inwardly by grace. (ST II II, Q6, A1)

Notice that Thomas ultimately attributes man's assent to God's grace. That is to say that man assents because God moves him by grace. Strange, no? Although we tend to think of man and God as two independent agents who compete with each other for space and power, Thomas explicitly rejects this understanding of God. Let's look at what he says in his writings on free will.

First let's look at Thomas' summary of the basic natural counterargument to the above:

Objection 3. Further, what is "free is cause of itself," as the Philosopher says (Metaph. i, 2). Therefore what is moved by another is not free. But God moves the will, for it is written (Proverbs 21:1): "The heart of the king is in the hand of the Lord; whithersoever He will He shall turn it" and (Philippians 2:13): "It is God Who worketh in you both to will and to accomplish." Therefore man has not free-will.

How does he answer this objection?

Reply to Objection 3. Free-will is the cause of its own movement, because by his free-will man moves himself to act. But it does not of necessity belong to liberty that what is free should be the first cause of itself, as neither for one thing to be cause of another need it be the first cause. God, therefore, is the first cause, Who moves causes both natural and voluntary. And just as by moving natural causes He does not prevent their acts being natural, so by moving voluntary causes He does not deprive their actions of being voluntary: but rather is He the cause of this very thing in them; for He operates in each thing according to its own nature. (ST I, Q83, A1)

Thomas says the same thing in other places, such as when he talks about grace. Basically Thomas disagrees as soon as someone says, "If God did it, then I didn't do it. And if I did it, then God didn't do it." Thomas usually speaks about this in terms of first and second causes. This means that God is a kind of transcendent first cause, altogether different from any reality we might encounter in the creaturely realm. God moves everything according to its nature, whether it be rocks, plants, animals, humans, angels, or anything else which exists. In him we "live and move and have our being" (Acts 17:28). Everything we have is from God, including our free will. God causes the existence and exercise of our free will (See: Divine Conservation vs Divine Concurrentism). Nothing at all happens without God, not even the use of faculties which he has bestowed. Yet this does not mean that our will is coerced or determined (Quaestiones Disputatae de Malo, Q6, A1). The Ground of Being is not in competition with created beings.

Summarizing Thomas

What does this mean? In short it means that God's causal activity does not preclude free will. If we follow Thomas and say that God gives the gift of faith and the grace that causes man to assent, this does not mean that man's free will is not involved in the assent. The assent comes from man. It also comes from God. To use the scholastic language, it comes from God as first cause and man as second cause. There is a mystery here, but the mystery centers on the nature of God as transcendent cause. What's important with respect to your question is that the atheist's free will is still at work.

Dumbing it down: Sufficient Grace

Let's take a moment to move away from Thomas down to a simpler level of analysis, to the (sometimes dreaded) concept of sufficient grace. Basically the Catholic Church says that God gives all men sufficient grace to have faith and be saved. Off the top of my head, two times this issue came up was around the 4th-6th centuries (Augustine to the Second Council of Orange), and the Jansenism of the 17th and 18th centuries. Since I don't have any resources with me other than the internet, let's just look at the first three Papal condemnations of Jansenism (many of which also relate to Calvinism):

1. that there are some commands of God which just persons cannot keep, no matter how hard they wish and strive, and they are not given the grace to enable them to keep these commands;
2. that it is impossible for fallen persons to resist sovereign grace;
3. that it is possible for human beings who lack free will to merit; (Jansenism - Wikipedia)

(Remember that these are condemned propositions that are attributed to the Jansenists. According to the Catholic Church, they are three false propositions. They are not the Catholic Church's positive teaching, they are something She condemns.)

The most relevant to your atheist is #1. God doesn't command the impossible; he always gives the grace necessary to fulfill his commands. In Catholic tradition this grace is referred to as sufficient grace. That is to say, he gives sufficient grace for the atheist to assent and have faith. If the atheist does not have faith, it is not because God did not give him sufficient grace to do so. God always gives sufficient grace.

(Remember that this is a different level of analysis than that of Thomas Aquinas above. Here we are not probing into the mysterious nature of God's causality and trying to understand the exact way in which he relates to man's free will. Here we are just looking at Catholic doctrine on the question, which is not going to go into the subtle details that Thomas [and his successors] address.)

The Kindergarten Answer

Let's drop down one more level. Children in Catholic schools would be taught that God offers everyone the gift of faith and each person gets to decide whether or not to accept that gift. Professional theologians might cringe at this explanation, but it's arguably not as far from the truth as some like to claim. Since you quoted Thomas on a difficult matter, I tried to give the high-level explanation, but for most people the kindergarten answer would suffice.

(According to Catholicism it might be said that theologians like Jansen and Calvin ought to have just stopped at this elementary level instead of moving on to the theological calculus, becoming confused, and erring.)

By choosing to turn one's back on God, one condemns oneself ...

CCC 1033 To die in mortal sin without repenting and accepting God's merciful love means remaining separated from him forever by our own free choice. This state of definitive self-exclusion from communion with God and the blessed is called "hell."

An atheist does not repent (repentance ~ to turn toward God, or to return to God), nor accept God's love (how can one accept love from a person whom you insist does not exist?) and therefore remains separated from God forever by their own free choice. Unlike the Calvinists, the Catholic Church believes in free will. The willfulness figures into the self-condemnation, and so the asserted blamelessness in the question has no standing in Catholic belief.

• Catholics do not subscribe to "once saved always saved." Even with the gift of faith, from God, and the remission of all sins at baptism the Catholic Church teaches that we are all sinners and that we can screw it up. (CCC 1704 - 1709¹). The sacrament of penance and reconciliation has a purpose: to restore the state of being in communion with God. The question has an embedded assumption of a lack of agency on the part of the atheist. That is a Determinist assumption that is not valid for Catholic belief which embodies free will.

While there is more to this regarding particular judgment, which I covered here, in a nutshell God doesn't condemn the atheist: the atheist condemns himself, or herself. To use a bit of imagery: the hand of God is there, extended, in the eternal offer of friendship and communion.

An atheist is not ignorant of God. (An agnostic could make that claim). An atheist rejects God. In theological terms, that is a willful turning away when one considers Catholic teaching. The offer remains there, even with the face and heart turned away from God.

This is God's will (as stated in the CCC):

CCC (Prologue) 2: God our Saviour desires all men to be saved and to come to the knowledge of the truth. (ref 1 Tim 2:3-4.)

The grace that God offers aligns with the point in your question. I want you to come to me is God's offer. The atheist (or anyone for that matter) may accept or not accept that offer. The acceptance is an exercise of free will in Catholic belief.

From the opening statement of belief in the CCC:

I. The life of man - to know and love God
CCC 1 God, infinitely perfect and blessed in himself, in a plan of sheer goodness freely created man to make him share in his own blessed life. For this reason, at every time and in every place, God draws close to man. He calls man to seek him, to know him, to love him with all his strength. He calls together all men, scattered and divided by sin, into the unity of his family, the Church. To accomplish this, when the fullness of time had come, God sent his Son as Redeemer and Saviour. In his Son and through him, he invites men to become, in the Holy Spirit, his adopted children and thus heirs of his blessed life.

The atheist rejects this call, and this gift, freely.

CCC 1022 Each man receives his eternal retribution in his immortal soul at the very moment of his death, in a particular judgment that refers his life to Christ: either entrance into the blessedness of heaven-through a purification or immediately, -- or immediate and everlasting damnation.

This was discussed in some detail in this answer here.

What is Grace?

Grace is an undeserved gift from God.

CCC 1996 Our justification comes from the grace of God. Grace is favor, the free and undeserved help that God gives us to respond to his call to become children of God, adoptive sons, partakers of the divine nature and of eternal life. [cf. John 1:12-18; 17:3; Romans 8:14-17; 2 Peter 1:3-4.]

You asked about faith as being gifted, assisted, even inspired by Grace. If one rejects the gift, it is by an action of the will: as above, we are back to free will.

¹ Some teaching on Free Will ...

CCC 1704 The human person participates in the light and power of the divine Spirit. By his reason, he is capable of understanding the order of things established by the Creator. By free will, he is capable of directing himself toward his true good. He finds his perfection "in seeking and loving what is true and good."7

1705 By virtue of his soul and his spiritual powers of intellect and will, man is endowed with freedom, an "outstanding manifestation of the divine image."8

1706 By his reason, man recognizes the voice of God which urges him "to do what is good and avoid what is evil."9 Everyone is obliged to follow this law, which makes itself heard in conscience and is fulfilled in the love of God and of neighbor. Living a moral life bears witness to the dignity of the person.

1707 "Man, enticed by the Evil One, abused his freedom at the very beginning of history."10 He succumbed to temptation and did what was evil. He still desires the good, but his nature bears the wound of original sin. He is now inclined to evil and subject to error: Man is divided in himself. As a result, the whole life of men, both individual and social, shows itself to be a struggle, and a dramatic one, between good and evil, between light and darkness.11

References: (Gaudium et spes)
7 GS 15 #2; 8 GS 17; 9, GS 16; 10 GS 13 #1; 11 GS 13 #2.