My ex and I got divorced without child support being decided through the courts. I moved from Israel to the USA. She cut contact with me and I stopped payment of child support. Now, a year later she just emailed me telling me to pay her child support. She is remarried, living in Israel. Do I owe child support according to Israel laws or the state I reside in or not at all because she remarried and the child's father lives abroad ?

You are liable to support your child, even though you live abroad permanently and under rules of private international law your obligation is according to the laws of Israel, because that is where the child lives, and his needs are to be met, and can be best assessed, and the dispute heard. You cannot escape that financial obligation because your child's mother has remarried, and you do not live in Israel. 

As you are not in contact with your child either, it would seen advisable, and in the child's best interests,  to either enter into negotiations to reach an agreement to deal with both issues i.e. the re-establishment of contact and the rebuilding of your relationship with your child, including virtual contact and meetings, in Israel and overseas, and the issue of child support. This can be done via direct negotiations between lawyers, or via opening a family dispute file at the Family Court, although this is not practical, if you are not intending to visit Israel to attend at least one meeting, in which case you can initiate matters, by applying for contact, or she can, by applying for child support, and shorten the default  pre-court procedure for alternative dispute resolution, which came into force in July 2016.

Our office represents fathers resident overseas in such negotiatons and it is possible to reach an agreement that can be authorized in court without you having to attend in person, if necessary.