I think you'd have a hard time convincing non-Notre Dame people that Notre Dame is in quite the same league academically as any of the Ivies beyond perhaps Cornell.
In my (admittedly biased) opinion, Notre Dame is comparable for undergrads to Columbia and Penn, but lacks the med school and maybe high-end grad programs.
Brown's smaller and completely liberal arts-focused, so not really a relevant comp. Dartmouth is similar culturally in some ways to Notre Dame but is probably more selective in admissions, in part because it's a lot smaller.
Harvard, Princeton and Yale are at a higher level. Among the five or six best universities in the country (with MIT, Stanford, Duke). I think it's OK to acknowledge that, and aspire to develop what they've got.
What all this has to do with the current state of college basketball, I'm not entirely sure.
Depends on how you define an 'academic league.' ND applicants are self-selecting to the point where we don't get as many, but our student body has about the same credentials (ECs, SAT scores, GPA, etc.) as any school in the country. Without the mass appeal that secular institutions have, our acceptance rates are higher.
If you're just looking at undergrad education, a lot of top schools derive their reputation from graduate and medical programs. For example, Harvard is well-known for considering undergrads a 'necessary evil.' That's why I didn't list them as a good example. I'd say we're already pretty much on par with Duke, maybe a little better, but that's immaterial. At a certain point, the differences between schools in terms of academic quality mean far less than the differences in focus and culture.
Notre Dame has always striven for holistic education. That means school, faith, and sports (mind, spirit, body+leadership). That's a valuable difference from a lot of schools that seem to focus on education as narrow skill development (cough cough MIT). Obviously, there's room to improve and I have a lot of ideas on that. However, that improvement doesn't mean our focus should be on 'becoming as good as Harvard' (whatever that means) but on becoming a better Notre Dame.
Edit: I'd like to add that a good fraction of what ND is today is a result of the anti-Catholic policies of the the Ivy League, among others. That kind of discrimination isn't OK, and is not part of the value set to which we should aspire, regardless of society's general view that it's alright to be bigoted as long as it's against Catholics.