Tell him you don't want to wait any longer and ask him for names of other trainers with his level of experience in working with back injury clients. He'll either schedule with you or he won't.
If your convinced it's something with you, rather than him, my only guess could be given the information provided is that you sound as if maybe you've put him on a pedestal and maybe are expecting things from sessions that he doesn't have to offer. Perhaps he senses that? Though he should just tell you that. So, who knows.
There's no way to know another's motivations, or lack thereof. You asked, he gave you his answer - which is for you to sit around waiting for him to call. Oh how typical!!
He either doesn't want to work with you, or, you shouldn't want to work with him! A month is far, far too long to wait to set up a session. If he doesn't want to work with you, well, the reasons don't matter.