Does my son's father have any grounds to take me back to court to gain visitation rights?

I have 2 sons. Their father has not seen or spoken to them in over 3 years. We had temporary orders during the first year of this absence drawn up that outlined a reintroduction process, which he never started. For our final hearing a year later, he voluntarily waved all visitation and contact rights. He is not allowed to contact the children by any means. This was his choosing and the language reflects this. He is now remarried and has stated he wishes to see the kids, but has so far not followed through with the reintroduction phase and is now sending messages through their cousins at school, as well as withholding medical reimbursements. He is required by law to pay child support, provide medical and dental insurance and reimburse half of all non-covered expenses. He is now refusing to pay any expenses without detailed billing information, which he actually gets through the EOBs as he is the policy holder.