You perhaps could use biology to rule out certain external characteristics of the human soul, but you can’t disprove the existence of the soul or afterlife. It would be an academically dishonest argument.
The exact nature of the human soul is undefined by church teaching, however both biology and theology can inform one’s one opinion as to the nature of the soul. However, any honest argument must admit that the existence of the soul is ultimately a matter of faith. For faith to be relevant, the God it is placed in must exist. For a Catholic, it is axiomatic that God exists. As such, petty appeals to biology are wholly unconvincing.
To a Catholic, faith and reason are not in conflict. Scientific inquiry is not to be shunned, but its results must be interpreted within the bounds of its discipline. In this sample problem, all we have shown is that consciousness cannot be measured when there is no brain activity to measure. It is simply dishonest to say this alone disproves the existence of the soul or of God.
This is not a discussion about what we can prove or disprove. Like the existence of God, the existence of an afterlife or a soul is an unfalsifiable proposition. Therefore, proving or disproving them is already a non-starter. What we are talking about is whether we can take our working knowledge of the Universe as it stands to inform these type of questions.
The idea that science can only operate within its own, various, narrowly defined parameters is demonstrably false, I’d argue. Discoveries in Chemistry, for example, can easily inform questions of biology, psychology, even astronomy. To say nothing of realms of knowledge that aren’t even traditionally scientific. What we learn about biology can change a former understanding of history. We can diagnose, with reasonable accuracy, the disease that killed someone centuries ago, if only by reading the scribblings of contemporaries who might have been prescribing leaches and spirits as cures.
So, I see no reason why the claims that theology makes (however unfalsifiable) cannot also be evaluated in the lights of our scientific understanding of the Universe. What neuroscience has long told us is that everything that distinguishes you from an organic bag of muscle, bone and water; everything that makes you
you, in other words - everything can be damaged or destroyed by damaging the brain. Conversely, what theology would have us believe is that when you die, and your brain is damaged completely and irrevocably, everything that makes you you will survive, fully formed and without impediment - and if anything, better than it ever was.
Though not rising to the level of proof, what we know about consciousness from neuroscience gives us good reasons to doubt the claims of theology re: an eternal soul and an afterlife - claims that, frankly, didn’t offer much of anything in the way of evidence to support them in the first place.