Caesar Cardini was an Italian American who set up restaurants in Tijuana to accommodate booze tourism during prohibition, just on the other side of the U.S.-Mexico border. It was in one of the Tijuana locations where he (or possibly his brother) first developed the Caesar salad. When prohibition ended, Cardini shut down his businesses in Mexico and continued operating exclusively in the U.S.

So really, all 3 countries have some claim to the salad: Italy, The United States, and Mexico.