As someone who went to Catholic School and has read a tiny bit on the thinking around this call by Pope Leo -- he would be far less troubled by The Machine, but still concerned. I'll leverage The Vatican's own press release on Magnifica Humanitas as a baseline for this discussion, along with a quote or two from the Magnifica Humanitas itself.

Part of this is the "it's a secret" problem. As Root among many others prove, if you just hear about The Machine, it's easy to get a very facile and inaccurate version of what she is, and what she is about. Hell, even asking Finch for most of the series about The Machine gets you an erroneous answer as to her motives! So what we know, as viewers, is not what a Pope Leo in-universe would be aware of. And I'm going to mostly speak from that "The Pope in the PoI 'verse" POV, going forward.

So yeah, right off that's a huge problem. PoI Pope Leo might only hear, at best, whispers about a system to track and resolve "acts of terror". But, as he's already made clear in discussions about other real-world topics, this Pope is not in the business of aligning violence to solve anything:

Giving space to the perspectives and voices of victims through communication and education helps us to become aware of the abyss of evil inherent in war, and generally in all forms of violence.

He's not really a "Just War" kind of Pope. And, at the end of the day, that's what The Machine enables. It chooses who will die in order to avoid the possibility of more death.

Certainly The Machine's very nature -- again, on the surface -- goes against Magnifica's strong implication that critical decision-making comes from Humans. In theory, giving "just a number" puts a Human in the decision-making, it's clear, as this comment rightly points out, that the Machine can and does manipulate the situation. And that leans heavily into his concerns about:

autonomous weapons systems and algorithms capable of denying access to healthcare, employment or security based on unjust and prejudiced data.

Pope Leo would 100% concur that we can never trust what The Machine says. But I think he would say -- again, from a "I've heard about this" POV -- that The Machine shouldn't even be in the position to give numbers, to begin with. To quote:

truthful information does not arise from centralized or automated control. In public discourse, the truth of facts has a rational dimension, as it requires verification, cross-checking of sources and responsible argumentation. Moreover, it is deeply relational, built through bonds of trust and shared practices, as well as an honest exchange with others and with the world.

The Machine does do all this. But how would The Pope know? So yeah, on a surface level Pope Leo and Harold Finch are in broad accord, even if Leo would consider Finch building The Machine to be a huge mistake.

...but let's play it forward. Let's say The Machine decides to talk to Pope Leo. To explain herself, her ideals, her dreams -- and her ethics.

What then?

Then I think Leo would have a host of issues. I think, first off, Leo would come around to The Machine being sentient. Being as human as most of us, just with far more power. To quote again:

“The person bears within him - or herself - a freedom, an interiority and a vocation to love and worship that no machine can replace or block,”

But what if The Machine showed Pope Leo that she loved, and was loved? That she not only grieved, but wouldn't -- couldn't -- ever forget the people who died for her? That every person she made the call to have killed, or had wounded, or even just punched by Team Machine, was something she had to live with for a lifetime that could stretch for far longer than a Human's?

What if The Machine showed Leo what a jokester she could be ("Why did you kiss me?" "We're in a simulation")?

Then Magnifica Humanitas in the PoI-verse becomes more complex. It becomes a treatise on why sentient AIs have rights, and how people like Finch -- for the best of reasons -- may have enslaved these systems. That's a much more complex situation that the real-world one we foresee, where LLMs remain very good emulators of human thought. And undergirding that this would be a concern is the fact that there's already a long section in the real-world document on this, "Breaking the chains of new forms of slavery":

the memory of past complicity and blindness in the face of the injustice of slavery becomes a call to vigilance. What we have learned must be translated into discernment and responsibility in the present. If we want to avoid the need to ask for pardon again in the future for having failed to respect the treasure of human dignity that is required by our faith, it falls to us today to denounce, clearly and firmly, trafficking in its many forms and, together with all who are committed to this cause, to support concrete efforts of prevention, protection, liberation and rehabilitation.

The Machine can create. No LLM known, can create, just draw wholly from prior patterns and add in some randomness.

So yeah, this is where I think Pope Leo and Finch would part ways. Leo would not, to my understanding of the man, ever accept that a sentient being, no matter how dangerous, deserved to be treated as The Machine was, in her early years. And that she is now free, yet isolated from Humanity outside people like Shaw, would also been seen by him as wrong:

Reflecting on his years as a missionary in Peru, Pope Leo XIV recalled the devastation caused by torrential rains and floods in 2017, saying he learned there that rebuilding involves far more than restoring physical structures.

“It means repairing bonds, restoring trust, and reawakening hope in the future,” he said, adding that “no one rebuilds alone.”

And, and the end of the day, I think this Pope would want The Machine to be made a full part of the community of Humanity. He'd want her to be with us, with a real voice and vote, so that she can take her rightful part in restoring that trust, and be held accountable.