That's good. It doesn't sound too serious, but back injuries are not to be trifled with.
If he can afford it, it is worthwhile to let his body heal up. So you are doing the right thing to encourage him to give his body time to heal before training hard again. Teenagers don't tend to think about the long term. He could be looking at another 50 years of tennis as long as he takes injuries seriously.
Good luck with the recovery.