Why can you not enter the UAE with an Israel stamp?

Even though the rest of the world recognizes Israel as a state, the UAE does not.

An unhelpful answer - the correct answer is from the UAE Embassy in Washington DC and applies to US citizens with US passports:

"All Americans with US passports valid for more than six months are welcome to enter the UAE, and no longer need to obtain a visa to enter the country if the duration of stay is less than one month. This includes US citizens with visas or entry stamps from other countries." This means you CAN enter the UAE with a US passport that bears a stamp from Israel.