The darkest periods of western culture came during the periods where Church and State were inherently intertwined. Moving away from that was the best thing that ever happen to Western Civilisation.
Your language here suggests a sympathy for the old "dark ages" caricature of pseudo-history, which is -- as a term, and as a reading of the middle ages -- no longer taken seriously anywhere n historical scholarship, history of science, etcetera. It's not worth rehashing all of that debate, but essentially it is widely recognized today that the middle ages were a time of great achievements scientifically and otherwise.
Anyhow, it's also a fantasy there can be some truly "secular" foundation for the State which is not itself grounded in some religious or pseudo-religious first principles on the dignity of man, which undergird its entire legitimacy. Nowhere in reasoning from nature as it presents itself do we arrive at a conception that each person must be taken as a moral end in themselves, that enacting genocide is morally bad in all cases rather than merely bad for utilitarian calculations, and so on. And every attempt to derive these moral prohibitions from social contract theories cannot even explain them as wrong but simply disadvantageous; not to mention that the idea of a contract itself is only a retroactive, fictional reconstruction of the origins of society, so that the strength and usefulness of that fiction is immediately null when real social conflict erupts.
If you raise abortion to a first principle and a right (truly though, calling it a "right" to have a technologically armed surgical infrastructure subsidized and at the ready to dismember natural processes within your body is pure nonsense -- surely another term is needed at the very least), then you must have some first moral principle upon which that depends. Is it that individual choice must never be limited? We restrict choices in countless places in the modern State when something is deemed dangerous or destructive of dignity or life. Would you try to call it healthcare? We're discussing cases where the mother is not in danger, so calling this healthcare is absurd -- healthcare is ethically limited to fixing something when the body is ill or broken, and cases of surgically modifying healthy processes for elective reasons must fall outside it. If you call it essential healthcare, realize you will have to do the same for a woman one day who wishes -- from the privacy of her own home -- to apply genetic engineering modifications to her developing offspring at will using a kit that is barely sci-fi at this point; that's "her body, her choice" too under that principle, which is why that principle can never logically apply when we're not talking about something within her body but about a dramatic surgical or technological reversal of her body, which can certainly be legally restricted or outlawed. When there's nothing wrong and it's a healthy, normal process that is to be destroyed or reverse engineered, healthcare isn't the right word, and can never be the right word for surgically destroying or altering the body's functions for elective reasons.