This historian is notorious for his anti-Catholic agenda and even the Wikipedia (no friend to Catholicism) article on this book indicates how inaccurate it was. For example:

Historian of science Lawrence Principe writes, "No serious historians of science or of the science-religion issue today maintain the warfare thesis...The origins of the warfare thesis lie in the late 19th century, specifically in the work of two men - John William Draper and Andrew Dickson White. These men had specific political purposes in mind when arguing their case, and the historical foundations of their work are unreliable." ^[2]
Principe goes on to write "Despite appearances, White’s arguments are scarcely any better than Draper’s. White uses fallacious arguments and suspect or bogus sources. His methodological errors are collectivism (the unwarrantable extension of an individual’s views to represent that of some larger group of which he is a part), a lack of critical judgement about sources, argument by ridicule and assertion, failure to check primary sources, and quoting selectively and out of context. White popularized the baseless notions that before Columbus and Magellan, the world was thought to be flat and that the Earth’s sphericity was officially opposed by the Church.

more at https://en.wikipedia.org/wiki/A_History_of_the_Warfare_of_Science_with_Theology_in_Christendom

Treating this book like legitimate history is, in effect, aiding the enemies of the Church to attack her.

I agree this historian is anti-Catholic, yet he makes clear he is believes in the globe. At the same time, he docuмents what Catholics teach (flat earth) in order (he thinks) to make Catholics look stupid and against science.  ADW hates Catholic teaching about flat earth. I also agree there is no warfare between Catholicism and science.  But then, that leaves us with Catholic Fathers and Scripture who taught a flat earth, against the theory of the globe.  

It leaves us with a historian who lied in order to promote his agenda.  The original sources do not back up the claims of the historian.
Here is Jeffrey Burton Russell on the topic:

In my research, I looked to see how old the idea was that medieval Christians believed the earth was flat. I obviously did not find it among medieval Christians. Nor among anti-Catholic Protestant reformers. Nor in Copernicus or Galileo or their followers, who had to demonstrate the superiority of a heliocentric system, but not of a spherical earth. I was sure I would find it among the eighteenth-century philosophes, among all their vitriolic sneers at Christianity, but not a word. I am still amazed at where it first appears.

No one before the 1830s believed that medieval people thought that the earth was flat.

The idea was established, almost contemporaneously, by a Frenchman and an American, between whom I have not been able to establish a connection, though they were both in Paris at the same time. One was Antoine-Jean Letronne (1787-1848), an academic of strong antireligious prejudices who had studied both geography and patristics and who cleverly drew upon both to misrepresent the church fathers and their medieval successors as believing in a flat earth, in his On the Cosmographical Ideas of the Church Fathers (1834). The American was no other than our beloved storyteller Washington Irving (1783-1859), who loved to write historical fiction under the guise of history. His misrepresentations of the history of early New York City and of the life of Washington were topped by his history of Christopher Columbus (1828). It was he who invented the indelible picture of the young Columbus, a "simple mariner," appearing before a dark crowd of benighted inquisitors and hooded theologians at a council of Salamanca, all of whom believed, according to Irving, that the earth was flat like a plate. Well, yes, there was a meeting at Salamanca in 1491, but Irving's version of it, to quote a distinguished modern historian of Columbus, was "pure moonshine. Washington Irving, scenting his opportunity for a picturesque and moving scene," created a fictitious account of this "nonexistent university council" and "let his imagination go completely...the whole story is misleading and mischievous nonsense."

But now, why did the false accounts of Letronne and Irving become melded and then, as early as the 1860s, begin to be served up in schools and in schoolbooks as the solemn truth?
The answer is that the falsehood about the spherical earth became a colorful and unforgettable part of a larger falsehood: the falsehood of the eternal war between science (good) and religion (bad) throughout Western history. This vast web of falsehood was invented and propagated by the influential historian John Draper (1811-1882) and many prestigious followers, such as Andrew Dickson White (1832-1918), the president of Cornell University, who made sure that the false account was perpetrated in texts, encyclopedias, and even allegedly serious scholarship, down to the present day. A lively current version of the lie can be found in Daniel Boorstin's The Discoverers, found in any bookshop or library.

The reason for promoting both the specific lie about the sphericity of the earth and the general lie that religion and science are in natural and eternal conflict in Western society, is to defend Darwinism. The answer is really only slightly more complicated than that bald statement. The flat-earth lie was ammunition against the creationists. The argument was simple and powerful, if not elegant: "Look how stupid these Christians are. They are always getting in the way of science and progress. These people who deny evolution today are exactly the same sort of people as those idiots who for at least a thousand years denied that the earth was round. How stupid can you get?"

But that is not the truth.
http://www.veritas-ucsb.org/library/russell/FlatEarth.html

I agree this historian is anti-Catholic, yet he makes clear he is believes in the globe. At the same time, he docuмents what Catholics teach (flat earth) in order (he thinks) to make Catholics look stupid and against science.  ADW hates Catholic teaching about flat earth. I also agree there is no warfare between Catholicism and science.  But then, that leaves us with Catholic Fathers and Scripture who taught a flat earth, against the theory of the globe.  

Regarding the Catholic Fathers and Scripture, which teach a flat earth against the theory of the globe (as you rightly mention above); it seems to me that the Fathers (and Scripture) were not so eager to separate God from His Creation, as the later and especially modern Catholic teachers do. And as the Atheists do as well.

That seems like the bottom line, really.

It was definitely the goal, to keep true to the Word. Nothing the heliocentrics believe ever explains anything about the globe by Scripture or the Fathers. It seems those who believe in the globe are there just to put up stops.

It was definitely the goal, to keep true to the Word. Nothing the heliocentrics believe ever explains anything about the globe by Scripture or the Fathers. It seems those who believe in the globe are there just to put up stops.