City diversity doesn’t necessarily impact the number of racial incidents at a specific university, or the feel of that particular university. Houston is a very racially diverse city - 26% non-Hispanic white, 24% Black, 44% Hispanic and 5% Asian. But so are Nashville (57%/28%/10%/3%) and Durham (38%/41%/14%/5%). There are lots of universities in relatively diverse cities that have been implicated in racial incidents. (It may improve quality of life when stepping foot off campus, though.)
The Greek hypothesis is an interesting one - Greek life does tend to be racially segregated at universities (especially Southern ones) so that may play a factor in the segregation of social life, too.
Rice is more ethnically/racially diverse than the other two, but not by a whole lot, and because of different balances of students - Vanderbilt and Duke both have larger proportions of black students, for example (probably because Durham/NC and Nashville/TN are both more traditionally and currently black cities/states). Rice has more Hispanic students (probably the same reasons). Vanderbilt has far fewer Asian students and fewer international students than either of the other two.