If we have an employee who has a non-work related injury, what is the law on not paying him for the time he is unable to work?

As a general matter, an employer does not have to pay an employee when he or she cannot work; therefore, in the absence of an employment contract or some insurance coverage providing for pay while disabled, you would seem to not have to pay this employee while he is unable to work. Indeed, unless your company is large enough to be covered by the Family and Medical Leave Act (FMLA) (at least 50 employees within a 75-mile radius) and the employee also qualifies under the act (worked for you full time for a year, more or less), and he takes time off pursuant to FMLA for his medical needs, you would not even seem to have to hold his job for him. (That's under federal law; some states have their own FMLA-like laws; even though those laws generally parallel FMLA, it would be advised to check your state's laws as well in this regard, to be certain of your obligations.)