Yes, Israel should be given full membership in NATO, because there is no reason to exclude them. Also, including Israel in NATO would make them feel like other counties are listening to their concerns. If Israel feels like they are a part of the world conversation, they will be more likely to engage in a conversation in reverse with other nations and be more responsive to their concerns.
Israel is the spoiled child that knows it can get away with whatever it wants because the United States will always rubber stamp any action it makes out of fear of pissing it off. When they start acting responsibly and with the least bit of compassion for others maybe they should be given membership, but their present day actions shouldn't be rewarded.