Imagine that, instead of the Spanish and Portuguese colonizing much of the Americas, the French did instead around the same time period. What would this look like?
Well, it certainly goes trough significantly not only weaken, but outright cripple Spain and Portugal in the early XVIth with at the latest, preventing Castille-Aragon to conquer Mexico and Peru (this is not particularly hard, mind you, ask
@EMT about it ) but also to be an Atlantic power which is significantly more difficult. Getting rid of Portugal is evenharder, and a Franco-Portuguese rivalry is likely to poison this TL to no ends.
Admitting we reached both, we could have more focus on
France Antarctique-equivalent. There are still issues.
-Neither Spain or Portugal were never, ever, going to accept or tolerate a French holding so close to their territories, especially with a strong Protestant presence. So either this colonization is Protestant-free, either you really need to deal with Portugal.
-You'd also need an incitative for colonizing the Antarctique in France itself. Basically you need avoid the perceived threat that french colonization would be an harbor for religious/political factionalism, namely Protestants. And probably no too much private efforts from remaining late feudal estates as Bourbons (maybe, this being said, a Valois Brittany as a semi-apanage would have a chance).
Both requires a much stronger France, possibly shorter (and less weakening for France) Italian Wars, and certainly no Wars of Religion, which probably implies a
survival of Henri II at the latest.
https://www.alternatehistory.com/forum/threads/henry-the-ii-jousting-accident.359271/#post-10986734