What verified, uniquely Christian principles support scientific approach?

There is a common faith to Biblical belief and secular belief, called the LOGOS.

Science is based on the LOGOS

"that the West is ultimately about reasoned inquiry in search of truth, and its way of searching for truth has been profoundly shaped by the manner in which Judaism and then Christianity affirmed the idea that God of the Bible is a rational being—the Logos. That understanding of God made all the difference, not least because it rescued the Greek and Roman world from the nonsense of pagan religion and fit well with emerging Hellenic insights into disciplines like mathematics and the natural sciences."

Source: “Without Logos, the West is over.” A conversation with Dr. Samuel Gregg

Consider what Pope Benedict said:

Violence is incompatible with the nature of God and the nature of the soul. "God", he says, "is not pleased by blood - and not acting reasonably (σὺν λόγω) is contrary to God's nature. Faith is born of the soul, not the body. Whoever would lead someone to faith needs the ability to speak well and to reason properly, without violence and threats... To convince a reasonable soul, one does not need a strong arm, or weapons of any kind, or any other means of threatening a person with death...".[4]

...

"In the beginning was the λόγος". This is the very word used by the emperor: God acts, σὺν λόγω, with logos. Logos means both reason and word - a reason which is creative and capable of self-communication, precisely as reason. John thus spoke the final word on the biblical concept of God, and in this word all the often toilsome and tortuous threads of biblical faith find their culmination and synthesis. In the beginning was the logos, and the logos is God, says the Evangelist. The encounter between the Biblical message and Greek thought did not happen by chance. The vision of Saint Paul, who saw the roads to Asia barred and in a dream saw a Macedonian man plead with him: "Come over to Macedonia and help us!" (cf. Acts 16:6-10) - this vision can be interpreted as a "distillation" of the intrinsic necessity of a rapprochement between Biblical faith and Greek inquiry.

Source: Faith, Reason and the University -- Memories and Reflections

What verified, uniquely Christian principles support scientific approach?

This is simply an attempt to answer a rather unique question as best as I can. I hope this helps as both science and faith should try to work together and not be isolated from the other and make unfounded interpretations of truth.

Pope John Paul II's Encyclical Letter Fides et Ratio may offer as glimpse into the mind of what the Catholic Church teaches on this.

Profoundly convinced that “whatever its source, truth is of the Holy Spirit” (omne verum a quocumque dicatur a Spiritu Sancto est). Saint Thomas was impartial in his love of truth. He sought truth wherever it might be found and gave consummate demonstration of its universality. In him, the Church's Magisterium has seen and recognized the passion for truth; and, precisely because it stays consistently within the horizon of universal, objective and transcendent truth, his thought scales “heights unthinkable to human intelligence”. Rightly, then, he may be called an “apostle of the truth”. Looking unreservedly to truth, the realism of Thomas could recognize the objectivity of truth and produce not merely a philosophy of “what seems to be” but a philosophy of “what is”.

The drama of the separation of faith and reason

45. With the rise of the first universities, theology came more directly into contact with other forms of learning and scientific research. Although they insisted upon the organic link between theology and philosophy, Saint Albert the Great and Saint Thomas were the first to recognize the autonomy which philosophy and the sciences needed if they were to perform well in their respective fields of research. From the late Medieval period onwards, however, the legitimate distinction between the two forms of learning became more and more a fateful separation. As a result of the exaggerated rationalism of certain thinkers, positions grew more radical and there emerged eventually a philosophy which was separate from and absolutely independent of the contents of faith. Another of the many consequences of this separation was an ever deeper mistrust with regard to reason itself. In a spirit both sceptical and agnostic, some began to voice a general mistrust, which led some to focus more on faith and others to deny its rationality altogether.

In short, what for Patristic and Medieval thought was in both theory and practice a profound unity, producing knowledge capable of reaching the highest forms of speculation, was destroyed by systems which espoused the cause of rational knowledge sundered from faith and meant to take the place of faith.

46. The more influential of these radical positions are well known and high in profile, especially in the history of the West. It is not too much to claim that the development of a good part of modern philosophy has seen it move further and further away from Christian Revelation, to the point of setting itself quite explicitly in opposition. This process reached its apogee in the last century. Some representatives of idealism sought in various ways to transform faith and its contents, even the mystery of the Death and Resurrection of Jesus, into dialectical structures which could be grasped by reason. Opposed to this kind of thinking were various forms of atheistic humanism, expressed in philosophical terms, which regarded faith as alienating and damaging to the development of a full rationality. They did not hesitate to present themselves as new religions serving as a basis for projects which, on the political and social plane, gave rise to totalitarian systems which have been disastrous for humanity.

In the field of scientific research, a positivistic mentality took hold which not only abandoned the Christian vision of the world, but more especially rejected every appeal to a metaphysical or moral vision. It follows that certain scientists, lacking any ethical point of reference, are in danger of putting at the centre of their concerns something other than the human person and the entirety of the person's life. Further still, some of these, sensing the opportunities of technological progress, seem to succumb not only to a market-based logic, but also to the temptation of a quasi-divine power over nature and even over the human being. - Fides et Ratio

Pope John Paul II goes on to state more on Science and Faith in Seminary Formation in his Message to the Reverend George V. Coyne, S.J., Director of the Vatican Observatory.

The three hundredth anniversary of the publication of Newton’s Philosophiae Naturalis Principia Mathematica provided an appropriate occasion for the Holy See to sponsor a Study Week that investigated the multiple relationships among theology, philosophy and the natural sciences. The man so honoured, Sir Isaac Newton, had himself devoted much of his life to these same issues, and his reflections upon them can be found throughout his major works, his unfinished manuscripts and his vast correspondence. The publication of your own papers from this Study Week, taking up again some of the same questions which this great genius explored, affords me the opportunity to thank you for the efforts you devoted to a subject of such paramount importance. The theme of your conference, “Our Knowledge of God and Nature: Physics, Philosophy and Theology”, is assuredly a crucial one for the contemporary world. Because of its importance, I should like to address some issues which the interactions among natural science, philosophy, and theology present to the Church and to human society in general.

Turning to the relationship between religion and science, there has been a definite, though still fragile and provisional, movement towards a new and more nuanced interchange. We have begun to talk to one another in deeper levels than before, and with greater openness towards one another’s perspectives. We have begun to search together for a more thorough understanding of one another’s disciplines, with their competencies and their limitations, and especially for areas of common ground. In doing so we have uncovered important questions which concern both of us, and which are vital to the larger human community we both serve. It is crucial that this common search based on critical openness and interchange should net only continue but also grow and deepen in its quality and scope.

By encouraging openness between the Church and the scientific communities, we are not envisioning a disciplinary unity between theology and science like that which exists within a given scientific field or within theology proper. As dialogue and common searching continue, there will be growth towards mutual understanding and a gradual uncovering of common concerns which will provide the basis for further research and discussion. Exactly what form that will take must be left to the future. What is important, as we have already stressed, is that the dialogue should continue and grow in depth and scope. In the process we must overcome every regressive tendency to a unilateral reductionism, to fear, and to self-imposed isolation. What is critically important is that each discipline should continue to enrich, nourish and challenge the other to be more fully what it can be and to contribute to our vision of who we are and who we are becoming.

Both religion and science must preserve their autonomy and their distinctiveness. Religion is not founded on science nor is science an extension of religion. Each should possess its own principles, its pattern of procedures, its diversities of interpretation and its own conclusions. Christianity possesses the source of its justification within itself and does not expect science to constitute its primary apologetic. Science must bear witness to its own worth. While each can and should support the other as distinct dimensions of a common human culture, neither ought to assume that it forms a necessary premise for the other. The unprecedented opportunity we have today is for a common interactive relationship in which each discipline retains its integrity and yet is radically open to the discoveries and insights of the other.

Contemporary developments in science challenge theology far more deeply than did the introduction of Aristotle into Western Europe in the thirteenth century. Yet these developments also offer to theology a potentially important resource. Just as Aristotelian philosophy, through the ministry of such great scholars as St Thomas Aquinas, ultimately came to shape some of the most profound expressions of theological doctrine, so can we not hope that the sciences of today, along with all forms of human knowing, may invigorate and inform those parts of the theological enterprise that bear on the relation of nature, humanity and God?

Can science also benefit from this interchange? It would seem that it should. For science develops best when its concepts and conclusions are integrated into the broader human culture and its concerns for ultimate meaning and value. Scientists cannot, therefore, hold themselves entirely aloof from the sorts of issues dealt with by philosophers and theologians. By devoting to these issues something of the energy and care they give to their research in science, they can help others realize more fully the human potentialities of their discoveries. They can also come to appreciate for themselves that these discoveries cannot be a genuine substitute for knowledge of the truly ultimate. Science can purify religion from error and superstition; religion can purify science from idolatry and false absolutes. Each can draw the other into a wider world, a world in which both can flourish. - Science and Faith in Seminary Formation

I can recall in one of my high school astronomy classes my teacher (priest) stated that the bible does not teach us how the heavens go, but rather how to go to Heaven!

Introduction

Since no one has really addressed your first example I want to offer an answer that does so.

Update 1 (2018-06-15): I will accept an answer that confirms or refutes the statement in example 1 above. That's a good starting point.

Example 1: One of the Christian thinkers (Thomas Acquinas?) stated that

1. God created the nature,
2. the nature works in an ordered way (i. e. is not chaotic), and
3. God does not micromanage most of the time (i. e. when he does, it's a miracle; at all other times nature works without divine intervention).

Thomas Aquinas is perhaps the foremost philosopher and theologian when it comes to reconciling science and theology, and although he makes many important and unique contributions, he is a synthesizer of an older tradition. This older tradition is embedded in the Christian worldview.

Christian Cosmology

First, let's revise your propositions:

1. God is the creator of everything that exists.
2. Mankind is created in God's image (Genesis 1:27).
3. Mankind is able to understand everything that exists (at least in principle).

How do (1) and (2) lead to (3)? Because we are created in God's image we have a share in his intellect and rationality. This affinity of man's mind with God's mind allows man to recognize the patterns that God makes use of in his creative act. Josef Pieper says that man recognizes what God cognizes. It's like God is the ultimate door-maker and he has bestowed man with the master key which opens all doors. The master key is man's rationality. Everything God creates we can understand, because our mind is made in the image of God's mind.

St. John even uses the Greek word 'logos' to describe the Second Person of the Trinity (John 1:1), a word which is related strongly to the idea of divine reason. Of course after the logos became incarnate Mary named him Jesus.

An Eternal Belief in Man's Ability to Understand

The conclusion is crucial for science, "3. Mankind is able to understand everything that exists." That doesn't mean man doesn't have to work at it, or that it will be quick and easy to understand creation, but the Christian has bulletproof reasons to believe that man has the ability to understand creation. This answers the crucial question in philosophy of science, "Is man's mind capable of understanding the natural world?" Christianity raises man's mind to a pinnacle so high it is even able to find resonance with God's own mind. From such a height it can survey all of creation. That is an absurdly optimistic view of man's rationality! Unfortunately today we often take such a view for granted.

Other Religions

What about other religions? Let's just take a quick look at polytheistic religion and monotheistic religion.

The cosmology of polytheistic religion is inherently chaotic. With pantheons of gods and a multiplicity of creative forces giving rise to the cosmos, the universe itself does not have a unified status. If creation is a consequence of Marduk's battle with the other gods, then we have no reason to believe that it is uniformly rational or even consistent for that matter. Nor do we have reason to believe that man's mind will be able to understand it. The polytheists do not have any reason to believe the conclusion noted above (#3).

Monotheistic religions which see rationality as a key component of God's essence and believe that man is created in God's image have better reasons to believe (3). Let's take the Abrahamic religions. Islam shows some promise but ultimately fails due to its theological Voluntarism. For Muslims, God's will trumps God's intellect. God does whatever he wants, he is not bound by intellect or reason. God's freedom and power are utmost in Islam. This is especially true in recent centuries, for there was a time around the turn of the first millenium when Arabic science flourished and outpaced the West. (We can thank Thomas Aquinas for helping Christians avoid this theological error.)

I've actually decided not to address Judaism since I don't know enough to make an informed assessment either way. Judaism may have the requisite ontological understanding for a strong scientific foundation. I am more interested in the argument which says that the scientific prowess of the West is due to Christianity, rather than the argument which says that Christianity is the only uniquely science-birthing religion. (Also note that your first example is simply not unique to Christianity.)

TL;DR

To summarize: the West was built on Christianity and Western science flourished because of the Christian worldview which provided such strong reasons to believe in the unfettered potential of man's mind. Christians believe that the rationality of man's mind is akin to the rationality of God's own mind, and therefore man has the innate capacity to understand those things God created. That means Everything!

I encourage the question poster and other interested parties to consult the works of Dr. Hugh Ross of “Reasons to Believe” who is an excellent source of information of questions in this exact field.
http://www.reasons.org/

Dr. Ross also has a number of lectures available on YouTube that covers areas exactly like this. https://www.bing.com/videos/search?q=youtube+hugh+ross+scientific+evidence+christianity&view=detail&mid=02D614B29F37D38C366402D614B29F37D38C3664&FORM=VIRE

Here’s a few things off the top of my head

1. The Bible says in

1 Thessalonians 5:21 21 "...test everything; hold fast what is good."

This scripture while generally applied spiritually speaking can and has been implied for advocating a method of epistemology that is essentially “the scientific method”.

2. There are certain Bible verses that speak of epistemology in a way that advocates some of the rules of logic. An example

James 3:11 Can both fresh water and salt water flow from the same spring?

This again is usually taken metaphorically or spiritually but is essentially in logic known as the “Law of Non Contradiction”

3. Jesus in the New Testament makes a number of statements that can be used as encouraging the development of Epistemology and Empirical reason when considered literally and beyond their immediate spiritual context. Some examples: “The parable of the Foolish Man and Wise man building their houses” in Matthew 7:24-27 (Things not built on truth will have unstable foundations and not last). Sayings about you will “know them” by their fruit in Matthew 7:16 (ministers and false ministers). This verse has been employed as encouraged a certain use of empirical evaluation of things based on “the results”.

4. I once again invite interested parties to watch some Hugh Ross lectures on You-tube and other social media sites. Hugh Ross in the past has lectured how Bible verses support such things as "The Space Time Theorem", "The Cosmic Expansion of the Universe", "Dark Energy" and many other Scientific laws and theories. Here is a short one and half minute video where Dr. Ross lays out the basic "Big Bang Cosmology" and mentions how on 4 or so basic points the Bible agrees with it. https://www.youtube.com/watch?v=-BaW2zJ4a2c

5. The Greek Term used for "Truth" (Aletheia) in the New Testament also is another thing that encourages scientific reasoning, observation and reality testing. If you consult both the Lexical Biblical meaning and the Greek Phisiophical definition that it was borrowed from besides relating moral truth, and Truth that comes from God via revelation, truth is actually defined as "reality", something "that is not hidden or concealed". This defintion, like the "Test Everything" passage quoted prior actually encourages a form of empirirism, concept validity and reality testing.

Aletheia (Ancient Greek: ἀλήθεια) is revolution or rising in philosophy. It was used in Ancient Greek philosophy and revived in the 20th century by Martin Heidegger. It is a Greek word variously translated as "unclosedness", "unconcealedness", "disclosure" or "truth". The literal meaning of the word ἀ–λήθεια is "the state of not being hidden; the state of being evident." It also means factuality or reality.[1] It is the opposite of lethe, which literally means "oblivion", "forgetfulness", or "concealment".[2]

https://en.wikipedia.org/wiki/Aletheia

Definition: truth, but not merely truth as spoken; truth of idea, reality, sincerity, truth in the moral sphere, divine truth revealed to man, straightforwardness.

225 alḗtheia (from 227 /alēthḗs, "true to fact") – properly, truth (true to fact), reality.

[In ancient Greek culture, 225 (alḗtheia) was synonymous for "reality" as the opposite of illusion, i.e. fact.]

http://biblehub.com/greek/225.htm