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It is not reasonable at all to refer to this as church support. Errors push themselves in slowly. It would have crept in by the excuse that it was the domain of science. There is no evidence to show the the magesterium teaching round earthism ever. There is a difference between what was taught in some universities by some professors, and what Rome was teaching.

The majority of Fathers believed the Earth to be flat.

It is important as to what most ordinary people thought. All the evidence shows they did not accept the globe. Here are two examples which stand out

http://flatearthtrads.forumga.net/t145-hereford-cathedral-map-of-the-world
http://flatearthtrads.forumga.net/t141-hieronymus-bosch-15th-century-painter-flat-earth-painting

The latter is from the 15th century.

If you are tempted to scoff, remember that on such an issue, these kind of things are regarded as important evidence to judge what the popular opinion was. Historians are like detectives and deal with whatever evidence they have, even if it is sparse.

The magisterium did not teach that the earth is a sphere (or flat).  Church controlled universities taught that the earth is a sphere.  Even if one does not call this support from the Church, there were no objections from the Church.  It was within the power of the Church to stop the teaching of spherical earth at the universities and yet the Church allowed it.  

The magisterium, however, has taught against the literalistic interpretations of Scripture that some flat-earthers use to support their belief.

You claim that flat earth was the majority view, but I have also seen claims that spherical earth was the majority view among the Fathers.  It would take quite a bit of effort to actually figure out which one was the majority and it does not seem worth doing.  Either way, it is clear that the Fathers were not unanimous, which means they were expressing personal opinions that we have no obligation to believe.

The Hereford Cathedral map is a T&O map.  Here is a Wikipedia article on these maps.

The T and O map represents only the one half of the spherical Earth.^[3] It was presumably considered a convenient projection of known-inhabited parts, the northern temperate half of the globe. It was then believed that no one could cross the torrid equatorial clime and reach the unknown lands on the other half of the globe. These imagined lands were called antipodes.^[3][4]

Bosch's work on the inner panels of the triptych is clearly in a fantasy style with symbolic imagery rather than realism.  There is no reason to think the outer panels are meant as a realistic portrayal.  The tradition of making 3 dimensional representation of the earth goes back to at least the late 15th century since some survive from this time:
 

The magisterium did not teach that the earth is a sphere (or flat).  Church controlled universities taught that the earth is a sphere.  Even if one does not call this support from the Church, there were no objections from the Church.  It was within the power of the Church to stop the teaching of spherical earth at the universities and yet the Church allowed it. 

The magisterium, however, has taught against the literalistic interpretations of Scripture that some flat-earthers use to support their belief.

You claim that flat earth was the majority view, but I have also seen claims that spherical earth was the majority view among the Fathers.  It would take quite a bit of effort to actually figure out which one was the majority and it does not seem worth doing.  Either way, it is clear that the Fathers were not unanimous, which means they were expressing personal opinions that we have no obligation to believe.

The Hereford Cathedral map is a T&O map.  Here is a Wikipedia article on these maps.
Bosch's work on the inner panels of the triptych is clearly in a fantasy style with symbolic imagery rather than realism.  There is no reason to think the outer panels are meant as a realistic portrayal.  The tradition of making 3 dimensional representation of the earth goes back to at least the late 15th century since some survive from this time:
 

on the universities question, I would only be repeating myself.

You may still be obliged to believe someday what the Fathers taught even if it was not unanimous.

For the literal interpretation of scripture refer back to St. Pius Xs biblical commission.

On the maps. Clearly wikipedia is an authority for you. Enough said....

Boschs work is both imaginative and yet how the world was fundamentally viewed. It is not meant to be realistic as a map, but realistic as to the foundations.

Here is the science that Jaynek ignores....

Now this is what I'm talking about when you make statements but refuse to provide data to back it up.  Flat earth and fixed earth are aspects of the same geocentric model.  Round and moving earth are aspects of the heliocentric model.  The two are mutually exclusive even if during the attempt to overthrow geocentrism the spherical earth was considered possible by some who at the same time, refused to believe the earth was moving.  The fact is, both spherical earth (a pagan religious belief for the perfect deity) and moving earth around the sun (pagan heliocentric god) both originate from pagan occult science.      

I have provided so many quotes that support my claims about this.  I cannot understand what you mean by saying that I do not provide data to back it up.  What exactly is it that you need to see?  Do you need quotes directly from Ptolemy and Aristotle that show they believe the earth is a sphere and is also the center?

Virtually nobody has believed in a flat, fixed earth since the sixth century.  The main geocentric model that competed with the heliocentrism of Copernicus, et al. was that of Ptolemy. (There was another model by Tycho Brahe in the mix, but it also had a spherical earth.) What you refer to as geocentrism - the Jєωιѕн/Babylonian cosmology - is not even included in the Wikipedia article on geocentrism.  It has a separate entry under "Biblical cosmology. 

There is overwhelming evidence that there are more than two mutually exclusive models.  I cannot even understand how you can make such obviously wrong claims.