I have been working on a device like this as a side project for some time. The concept is feasible. I could build a device tomorrow using off the shelf parts that would allow you to breathe underwater using a membrane. This device would use commercial oxygenator membranes to interface the sea water in a closed circuit rebreather. Using Henry's Law, oxygen will flow into the diver's air supply and CO2 will flow out, adjusting to ambient levels, which is nearly the same as the atmosphere (partial pressure) in most recreational diving environments.
However, the biggest problem to overcome before building a product you can sell is not technological. As some have mentioned, the amount of dissolved gasses in seawater is small. My partner (PhD in fluid mechanics) and I have calculated that a diver swimming along at an average rate would need a lot of membrane material to breathe properly underwater, even at 100% efficiency. I have a design using existing commercial membranes that can fit the required material into a device about the size of a small refrigerator. This is obviously too large to fit on a diver's back.
To make such a device portable, I envision making an underwater drone carrying the membrane module. It would follow the diver who would be attached by a hose and a tether. It could also be driven by the diver like an underwater jet ski. The diver would limited in his movements, but could stay underwater for an indefinite amount of time. It could also extend the maximum diving depth to 500 feet by allowing for a longer ascent time.
If anyone is interested in collaborating on this, email me.