Anyway, you can request that they don't stamp your passport in Israel and they'll do it on a slip of paper instead. I had mine stamped because I was due for a new passport anyway and had no plans to be in a Muslim country prior to getting it.
Israel itself doesn't ban anyone with visitor stamps from Muslim countries as far as I'm aware, but they'll pick up on it at immigration. They don't seem to care when people are coming in, but when you're leaving, you get scrutinised more closely. I had to answer the same questions with three different people when they saw that I'd been to Jordan and Malaysia - Israeli passport holders aren't even allowed into Malaysia at all - and it was a bit intense, but this was all done while queuing for immigration desks rather than being pulled aside.