Is fleece an alternative to cotton (sustainable or otherwise) -- the answer is mostly no due to the vastly different properties of cotton fabric and fleece fabric. An alternative is only an alternative if it can serve the same (or a very similar) purpose.
Cotton is a cellulosic fiber that is hydrophilic, breathable and doesn't have a lot of warming power due to low air retention. Cotton is also quite sturdy and very abrasion resistant when woven. Most cotton fabric is smooth and rather thin.
Fleece on the other hand is rather hydrophobic, very warming and doesn't have a good breathability. Fleece is not very abrasion resistant and prone to pilling. Also, fleece is always knitted, and thus structurally less sturdy than a woven cotton fabric. Fleece is also thick and and fluffy.
If you think about areas where cotton is used a lot: bedding, towels, jeans, t-shirts or shirts/blouses, in almost none of these application areas would a fleece fabric make any sense at all. So substituting the one with the other would not be feasible and thus the question if the substitution would be more sustainable is futile.