Melanin is a "macromolecule", full with aromatic rings and carboxylic acids. Usually, the weak organic acids (except of low MW) are less soluble in water when they are almost) fully protonated. When these acids are protonated, they are less polar than its ionized form.
Surface of larger molecule(Melanin) containing bulky groups gives less interaction to polar sites for bonding and interaction with polar molecule like water.
Maybe the question is not if its soluble else how many soluble is?
The synthetic melanin of Sigma has a solubility of 20 g/L in NH3OH 1 N, but if you use NaOH 1N to solubilizate the melanin and after that you neutralize the melanin keep on soluble.
Try this, although it is general: S. Hansen, S. Pedersen-Bjergaard, K. Rasmussen: Introduction to Pharmaceutical Chemical Analysis, Wiley & Sons, 2012, p.302-310
Or
M. Jones and S.A. Fleming, Organic Chemistry, Norton Science
I am assuming that any other organic chemistry handbook might be OK.
Less general, and not for melanin, tannins, etc., but for carboxylic acids this might be a good start point: A. Pobudkowska, U. Domańska, Study of pH-dependent drugs solubility in water. Chemical Industry and Chemical Engineering Quarterly 2014 Volume 20, Issue 1, Pages: 115-126, DOI 10.2298/CICEQ120531116P
An alternative might be if you can calculate pH dependent logP (logD, eg. with Marvin). You will see that not ionized versions have considerably higher logP than the ionized ones. Of course, your macromolecule may also form pH dependent associates which can affect the solubilities.