Assuming it managed to reach an altitude of 100 km (40-50 km is far more typical), that would allow it to see a distance of ~1 000 km. But to see water you need a distance of at least 2500 km.

Would it be rude of me to ask how you came to this conclusion?

Draw a line from the altitude of the balloon to the centre of Earth, assumed to be spherical.

Draw a line from the balloon to a point tangent on Earth.

Draw a line from the point the tangent line touches Earth to the centre.

This forms a right triangle, with one length r, and the other length r+h.

The angle at the centre of Earth can be found as:

a=arccos(r/(r+h)).

The length along the surface can be found as:

l=a*r = r*arccos(r/(r+h)).

Putting in r=6371 km, and h=100 km, this gives:

l=6371 km * arccos(6371/6471) = 6371 km * 0.176 = 1121 km.

So a balloon at an altitude of 100 km will only see a distance of 1121 km along the surface of Earth.

Quickly look at a map of Antarctica, like this one here:

https://upload.wikimedia.org/wikipedia/commons/c/c0/Antarctica.svgWe can see that other than seeing accross a small outcrop, Antartica is well over 2000 km wide. In fact, we see the entirety of Antartica is outside 80 degrees south.

10 degrees of latitude is given by 10*10000/90 = 1111.1 km, so 2222 km for crossing the 80 degrees. (approximately, if you want it more accurately, it would be 6371*10*pi/180 = 1111.9, so basically the same).

Additionally, you can go to Google Maps, and let it do the calculation for you, and find the shortest distance is roughly 2500 km.

So that wont be proof.

Regardless of if Earth is round or flat, you would expect to see a lot of ice.

It is only in the models were Antartica is a thin wall of ice you would expect to see the edge of Earth or something.