Toadfish, barring the reef species, are estuarine specialists... that is, they live primarily at the mouths of rivers, where the salinity changes throughout the day with tidal movements. That leads me to guess that yes, they can live in brackish water, although it would require a gradual reduction in salinity over a few days to avoid osmotic shock, assuming you caught this fish in full marine water.
However, I'm not entirely convinced that this is a Gulf toadfish. I'm leaning towards leopard or oyster toadfish. They can be very difficult to tell apart due to the ambiguity in their appearance and the variation in their patterning. They can lighten and darken dramatically in a matter of a few minutes.
I'm not sure there is anybody working with any of the western Atlantic toadfish species, but there is an Indo-Pacific species that turns up in the trade sometimes, usually under the misnomer "freshwater stonefish" or "freshwater lionfish". That being said, they are not members of the scorpionfish family, nor are they freshwater, but another species of estuarine toadfish. They require salinity in their water, and reportedly, are very difficult to get established eating anything other than live fish and shrimp. (Unusual, because I've caught quite a few toadfish using scraps of fish and clam, but I digress)
Due to their slow movements and obscure behavior, they do best in a species-only tank, and should be kept alone due to their cannibalistic tendencies. If you think you're ready to go ahead and keep one, go for it. But be ready to have basically an aquatic pet rock that is tough to feed and poops a LOT.
They also have some concealed spines in their dorsal and anal fins, and they pack a wallop. So I'd suggest not handling them.