My American-born wife and I are fighting over custody in Israel. I am very close to the children, but the social worker's report is in her favour. If she does get custody I am sure she will want to move back to the States with them. What can I do to prevent this?

Unless you consent, she will be forced to bring a legal action to get court permission in Israel. If she does, you can file defence pleadings, and argue that relocation will not be in the children's best interests, and will/could result in irreversible emotional damage to them. If you really suspect she will abduct the children, you can apply for an ex parte 'stop order' to prevent the children leaving Israel.
Moving Abroad