Does weak muscles in back have anything to do with sciatic pain?

No : Sciatica is due to general compression or irritation of one of the 5 spinal nerve roots that gives rise to each sciatic nerve it is not related to weak muscles.

Indirectly maybe: Directly unlikely. Indirectly, weak "stabilizing" core muscles may lead to the condition. As core stabilizing muscles of the spine/abdomen/pelvis become weak or dysfunctional, it can lead to pathological stresses on the spine vertebra and discs. This can/may in turn cause degenerative changes(ie disc protrusion or osteophytes) that will cause radiculopathy (which often is referred to as arthritis).