While Christianity didn't invent most of the methods of science, there is a plausible case that it nurtured the emergence of modern science (rather than opposing it), and helped it achieve a sustainable impact on the world
The idea that religion, in general, and Christianity, in particular, has always been in opposition to science is very easy to believe, especially in a world full of fundamentalists. But it is not a true reflection of the history of either religion or science; it clouds our ability to understand why modern science arose and became strong in a particular place and time. We will not get a good answer to the question if we allow current prejudices to cloud the analysis of what actually happened.
That the story is more complex than that should be obvious even from the much repeated story of how the Church tried to suppress Galileo's ideas. For a start, the ideas the church was supposed to be defending (Aristotle's philosophy and Ptolemy's astronomy) were not Christian ideas at all). And it is a little rich to credit Aristotle as a pioneer of science in this context, when his failure to give primacy to observing what actually happens was one of the key problems holding back the development of Greek science.
Spreading the credit for many new scientific ideas to other cultures is a fair reaction to Eurocentric thinking in the past. But it triggers a very important question: why did science and technology stall in those other places? We should give credit where credit is due, but in every other culture where new scientific ideas were developed, the progress of science stopped and failed to achieve constant improvement and a major impact on the world. There is something unusual about what happened in Europe where science eventually emerged in a form where progress has not yet stopped and where its impact on the world has transformed the world. This is where it is worth looking at actual history to see if there might be some useful insights about why this only happened there.
And, it is very important to note that this is a question about what happened historically. It isn't going to get a good answer if we apply current prejudices. Treating the church of 16th century Europe like a bunch of modern fundamentalists opposing evolution isn't a good way to do history.
Here are some observations leading to an alternative answer:
First, the idea that Christianity has consistently opposed science is historically false.
As far as I can tell the idea of religion and science in perpetual conflict goes back to Huxley and his defence of Darwinism. The traditional skeptics story is that god was dethroned by Darwin and opposed by the leadership of the church. But it is worth looking at the real history. For example, in his book Darwin's Forgotten Defenders, David Livingstone argued that many religious leaders – even the evangelical protestants – embraced Darwin. As he argues in the introduction:
...we ought also to revise the notion that science and religion are inevitably at odds...The findings I present challenge that assumption.
The historic evidence is not consistent with the idea that something inherent in Christianity opposes scientific progress even around evolution.
Secondly, there is something in the Christian tradition that actually fostered modern science.
This case was argued in the short but dense book by R Hooykaas, Religion and the Rise of Modern Science, where a very detailed analysis of history shows that many of the philosophical ideas the led to modern science, especially the primacy of observation and practical uses of science originated in Christian thinking. In the very elements that, when added to Greek thinking, make a combination somewhat like modern science with "getting your hands dirty" experiments (which the Greeks would have rejected).
To quote in part from a much more recent summary:
Second, religious considerations might provide some of the presuppositions for scientific investigation. Here I have in mind theologically-grounded ideas about the intelligibility of nature, manifested in the early modern period as the notion that there were divinely authored laws of nature. Third, and related to this, is the way in which theological positions may, and indeed have underpinned specific methods of investigation. Fourth, religious convictions may provide the motivations for some individuals to pursue the formal study of nature.
While Hooykaas is hard to summarise, part of his argument is that the environment and philosophy of many of great church leaders around the time of the Reformation was entirely sympathetic to observational, experimental and practical science. They believed that you could understand God via understanding Nature so making science respectable and approved of in a culture where that was by no means a foregone conclusion. Even Calvin didn't reject observations of the world even if they appeared to contradict the bible (in contrast to many modern fundamentalists). Many skeptics and opponents of religion have bought the idea that modern fundamentalism is typical of all religious belief in history. In actuality the philosophy of fundamentalism (and its creationist children) is a very modern idea (see James Barr's Fundamentalism for a summary).
And there is something about the nature of the Christian (and Jewish) bible that leads to diversity. Unlike the Koran, for example, the bible is a work of many authors and does not have monolithic style. The reformation is arguably the result of people looking back at their core sources and seeing things that contradict the authority of the church leadership. Hence the core source for the religion is actually a little subversive and fosters a questioning attitude and not absolute authority for an organisation. That fragmentation of the religious power base was probably important for the development of science and certainly for preventing any single authority stamping out ideas it didn't like (which is arguably why Arab and Chinese science faltered).
Thirdly, history shows that religion nurtured and protected the development of modern science before it became robust and sustainable on its own
This argument has been well summarised recently by Peter Harrison in his lecture Religion, The Royal Society, and the Rise of Science. He expresses his conclusions from the study of the actual interaction of science and religion in England thus:
we can conclude that while there may have been scientific ideas and practices without religion, there would not have been a longstanding and evolving scientific culture of the kind that emerged in the West without the support of religion.
He bases this on the way religion protected emerging ideas in science from outside attacks that might well have had the effect of stifling the emergence of "modern science". In particular, there were many groups in society opposed to the activity and practice of science but religious leaders and organisations often protected and nurtured the activities of scientists until they were strong.
It is worth reading the paper for the detail as it is not just a fascinating source of interesting history but is also a good read.
In summary
It is a big topic that I don't think I have yet done a good service to. But the key issue is that there is something peculiar about the runaway success of science and its emergence in western Europe and that this is not a consequence of a eurocentric perspective. The Chinese, the Indians, the Arabs and the Greeks all had important ideas that contributed to science, but progress stalled in all those cultures. Christianity is at least one factor that made a difference.
It does not require a willingness to defend religion to make this case: it just needs a little skepticism about what happened in history. Skeptics shouldn't buy myths about what actually happened just because they sound good: thats what the religious are supposed to do.
Update
A recently published book offers a better summary of the above arguments culled from several historians and philosophers of science but especially Fr Stanley Jaki, a Jesuit scientist, theologian and philosopher.
In summary the argument outlined there is (my emphasis):
Modern science was born in a particular culture, a Christian culture, and we can trace its antecedents backward all the way into the early Middle Ages. No other culture—Greek, Roman, Indian, Chinese, Egyptian, Babylonian, Islamic, African, Mayan—ever gave us anything like modern science. Therefore, the cause of modern science’s success must lie, ultimately, in Christianity, and the more we dig into the Christian origins of modern science, the farther back we find positive evidence for the sustained, sophisticated developments that underlay modern science.
The basic argument, backed by extensive and detailed historical evidence, is that a certain particular philosophical mindset is required for science to develop and grow. For example the very idea that the universe is comprehensible and there are things in it that are worth discovering by experiment. The core of the argument is outline in the quote below (my emphasis again):
I do not think, however, that I have even yet brought out the greatest contribution of medievalism to the formation of the scientific movement. I mean the inexpugnable belief that every detailed occurrence can be correlated with its antecedents in a perfectly definite manner, exemplifying general principles. Without this belief the incredible labours of scientists would be without hope. It is this instinctive conviction, vividly poised before the imagination, which is the motive power of research: that there is a secret, a secret which can be unveiled. How has this conviction been so vividly implanted on the European mind? When we compare this tone of thought in Europe with the attitude of other civilisations when left to themselves, there seems but one source for its origin. It must come from the medieval insistence on the rationality of God, conceived as with the personal energy of Jehovah and with the rationality of a Greek philosopher. Every detail was supervised and ordered: the search into nature could only result in the vindication of the faith in rationality. Remember that I am not talking of the explicit beliefs of a few individuals. What I mean is the impress on the European mind arising from the unquestioned faith of centuries. By this I mean the instinctive tone of thought and not a mere creed of words. In Asia, the conceptions of God were of a being who was either too arbitrary or too impersonal for such ideas to have much effect on instinctive habits of mind. Any definite occurrence might be due to the fiat of an irrational despot, or might issue from some impersonal, inscrutable origin of things. There was not the same confidence as in the intelligible rationality of a personal being. I am not arguing that the European trust in the scrutability of nature was logically justified even by its own theology. My only point is to understand how it arose. My explanation is that the faith in the possibility to science, generated antecedently to the development of modern scientific theory, is an unconscious derivative from medieval theology.
These overall arguments are based on detailed comparisons of how scientific ideas developed in many other societies, cultures and religions. The dominant fact is the failure of science to develop and expand to its modern form in any of those societies despite many of them having an apparently large lead on medieval Europe in some key technologies and scientific discoveries. Those who argue that Chinese, Arab or Greek ideas were the real origin of modern science need to find convincing explanations for why the development of science stopped in those cultures and failed to produce what we know as modern science.
The historic details are significant. One key development was made by the 14th century French Priest Buridan who was instrumental in breaking the Aristololean orthodoxy about the motion of objects and of being the first person to express the modern concept of inertia. Buridan opposed the theological implications of Aristotle's philosophy. But overthrowing that orthodoxy (for example the belief that heavy objects fell faster than light objects) was essential for allowing modern empirical physics to develop.
I won't include more details in this already long answer, but the significant point to emphasise is simple: the modern belief that christianity is implacably opposed to knowledge and progress in science is simply untrue given what we know about history. And a reasonable case can be made that science as we know it is dependent on the philosophical underpinnings that came, almost uniquely, with christianity.
