Simplest way to carbonate a kitchen faucet
I don't believe this is going to work very well with the set up you have described.  Carbonation of water requires that the CO2 dissolve into the water.  Assuming you get the pressure all worked out, the surface area between the gas and the water is going to be tiny.  If it worked at all, it would dissolve into the water line itself.  My guess is only a few ounces would be carbonated at most when you open the tap.

Another thing to consider is that the colder the water, the more CO2 it can dissolve.  This is why a hot beer is more likely to foam out and create too much head than a cold beer.  Here's a graph of that relationship:

source

I think you might be able to manage this with a tank, though.  Basically it would be a keg but you would need to have a water supply as well as CO2 for the water to refill through.  I'm not sure there's something off the shelf for this but reverse-osmosis systems have a tank so there might be a way to cobble together parts for to make it happen.  I'm not exactly sure how you would make it automatically refill.

You'd want to have some sort of check valve on the water line to prevent it from flowing back into the water supply.  The CO2 tank should have some sort of pressure control valve just like it would on a keg system.

This wikipedia page has some good info on this.  If 70 PSI is the right number for the water line, you would probably need to release the pressure from the CO2 in order to refill the tank.  The amount of carbonation could be tweaked by the pressure from the CO2 tank.  The higher the pressure, the more CO2 will dissolve.

One other consideration is that it the pressure is much higher than typical water pressure, the faucet might not be designed for that and/or it could shoot out too fast.