Some Arab countries have reasonably good diplomatic relations with Israel and visiting these countries (listed below) after a visit to Israel shouldn’t be a problem whatsoever. It might not surprise you that the two other countries that share a land border (but not a crossing point) with Israel, Lebanon and Syria, do NOT permit visitors to their country if any trace of a visit to Israel is found in a passport. If a second passport is out of the question, when arriving in Israel ask the immigration clerk to issue the Israel passport stamp on a separate piece of paper because you intend visiting Arab countries after your visit to Israel.