Yes, people from the Middle East are white, unless we are talking about someone transplanted who just happens to live in there and they are actually East Asian or black. For instance, we can’t say north Americans are all white, but the way we use the term Middle East currently, they would be white. My husband is Mexican and his national background is half Middle Eastern, quarter Spanish, quarter French. On the US census he is Mexican and Caucasian. The definitions for these things are kind of archaic now in my opinion.
If you are American, Asian means East Asian, which is the old Oriental (places like Japan, China, Vietnam, etc.). If you are talking about the continent of Asia, you would have to throw continent in there to clarify you mean the continent. It’s a matter of communicating well, rather than knowing geography well. The Middle East is in Asia, but no one uses Asian to describe a Middle Easterner. Not in America. Other countries might use the words differently.