I am a crazy person, in some views, because I believe completely that a Judeo-Christian (Catholic) worldview is critical to scientific inquiry as we know it. You have to believe in objective truth, you have to believe that creation is governed by precise and inalterable laws of nature. Such beliefs are completely unprovable, and many human civilizations have proceeded without them. The notion that most scientists are atheists is actually laughable, because even if a scientist thinks he/she is an atheist, the exploration of creation AS IF IT WERE BOUND by inalterable laws is itself a leap of faith. Why is this Judeo-Christian? Mainly because this is the only theology in which a unitary God is completely unchanging throughout eternity. Not only does this require monotheism (because multiple gods always quarrel), it actually excludes Islam in which Allah seems to have these changes of heart. In all these things Christianity completes the revelation given to the Jews - there is no other that does so.

Methodological atheism is a faith that there are physical laws governing the universe. The notion of physical laws is nowhere guaranteed by anything. You can observe evidence supporting them, you can believe them, but you cannot prove them. Stick a fork in it.

a) that Heaven as a whole is moved in its daily rotation around Earth by God
b) that heavenly bodies (great and small luminaries and presumably also invisible ones) are moved by angels.

Here is a Latin version of propositions condemned, chapters ordered thematically, so double numeration (number in chapter and number in original condemnation), with comments/footnotes also in Latin by me:

http://ppt.li/tempier

One of these vestiges you say is the autonomy of science from theology, which is a direct consequences of the 1277 condemnations.

After Charles Darwin clearly demonstrated that faith is not necessary for a scientific understanding of nature, it is better said as "Scientific progress requires a dialogue with faith"

A second way the Catholic faith can criticize "science" is when scientists, or people who claim they speak for science, or who say "science tells us X" claim that just because we can do something it is moral for us to do it.

If Catholics want me to applaud their faith for disputing such a claim, I don't mind doing so. Most of us secularists, though, think the absurdity of that claim is too obvious to need the endorsement of a religion or any other authority.

Science, on the basis of observation CONCLUDED the regularity of the universe.

Except that the belief in a rationally-ordered universe long predates anything we would call modern science. Alas, in history effects cannot come before their causes.
For example, Adelard of Bath in the 12th century, wrote in a dialogue with his nephew:
"[T]he natural order does not exist confusedly and without rational arrangement, and human reason should be listened to concerning those things it treats of. But when it completely fails, then the matter should be referred to God. Therefore, since we have not yet completely lost the use of our minds, let us return to reason."
-- Quaestiones naturales
and again:
"If we turn our back on the amazing rational beauty of the universe we live in, we should indeed deserve to be driven from it, like a guest unappreciative of the house into which he has been received."

George suggests the voluntarism propounded by al-Ghazali, which did indeed stifle science in the Islamic world. Ibn Rushd argued against it -- fire really does burn cloth, he insisted -- but he lost the intellectual battle there.
But if science would work better with Confucianism, one is stuck with the problem of why it did not. Needham, the great scholar of Chinese technology asserted just the opposite. (See comment above.)