How applicable are software engineering internships to computer science research and graduate school?
You are correct that research experience is extremely valuable for getting into graduate programs.  Internships are one way of gaining such experience, but far from the only way.  At a high-ranked university, however, there will also be lots of opportunities with professors, both during the term and in the summers, and those will be at least as applicable, if not more so, for getting into graduate school.  Industry experience as a full-time employee can be great for showing maturity and various skills not typically acquired during undergraduate, but frankly, you're not going to develop a significant and differentiating amount of such skills in a couple months as an intern either.

In short: no, you don't need industry experience to get into graduate school.