No, most of these people's lastnames actually originate from the English language and have nothing to do with the Asian origin.

Lee, here, is actually just another spelling form of the English original Laye ( there are even different spellings for that too), a word comming from middle English, meaning clearing or meadow. Another alternate spelling form is Leigh, which means the same, and can also be used as first or last name just like Lee. This is the reason why many families or european decent carry this last name, while having nothing to do with the asian last name Lee.

Now, that we think of that last name as exclusively Asian is actually more of mix of frequency of family or clan names sounding phonetically like Lee, they are very common, and the need to transcribe from the original respective language alphabet into our letters for international communication - Lee in this case is a transcription of the phonetics of the symbols in the different Asian language spelled out with our alphabet. Yi, I, Ee, Ri, Rhee, Rhie, Li and Lý, are all different forms of very common Chinese and Korean names that can be turned from their orginal Korean or Chinese symbol into the Lee we know in order to express the phonetic sound of the name in our alphabet, but they all also can be spelled like I did above.

Thus, there is indeed a difference between the name Lee of English decent and that of asian decent.