Is allspice traditional in the Levant?
Allspice is, indeed, the only commonly used spice* native to North America.  It is also very commonly used in various Arabic, Central Asian and Middle Eastern cuisines.  It is a standard part of the cuisine-defining North African spice mixes baharat and ras al hanout.  The vast majority of global Allspice is still grown in Central America.

You ask a good question, because I can't find any information about how it got to North Africa and the Middle East.  Allspice was original brought back to Europe by the Spanish and Portuguese as "Jamaican Pepper" or "myrtle pepper", back when they were still claiming that North America was China/India.

Given this, I have two theories about its introduction into those areas:


Allspice was traded to North African slave traders as part of the slave trade into the Americas in the 17th century.  From there, it spread through the Arabic-speaking world.
Allspice was traded to Indians (who also use it) in exchange for black pepper (just as capsicum peppers were) by the Portuguese.  From India, it spread through the Arabian trade routes of the Indian Ocean.


I don't have any documentary evidence towards either of those theories; my books on the history of spices largely omit allspice.

References:


https://iwp.uiowa.edu/silkroutes/allspice-jamaica-pepper
https://www.britannica.com/plant/allspice
http://theepicentre.com/spice/allspice-2/


(* spice, as opposed to herb or fruit)