I guess he has a point in a very narrow sense. But as Lord Opie pointed out, there are also societal benefits to your being on a bike, never mind the fact that you own a car too.
This argument, that only those who pay for a service should be able to use it, is kind of the flip side of the argument that only people who use a service should have to pay for it. Like home-schoolers who don't like paying property tax to send the neighbor's kid to public school. In both cases, the argument is that everything should be supported by user fees alone. In reality, much of our economy doesn't operate that way, and probably couldn't. So life's messy, get over it.
Besides, we could ride on unmaintained paths if we had to, could he? If there were no cars and no gasoline tax to pay for roads, there would have to be something else to pay for whatever mode of transportation people chose instead.
Ask him if he would support going to all toll roads, with a sliding fee according to size of vehicle and wear and tear on the road.