There's really two questions here. Who, legally owns the name "Rush," and what makes up the real soul and identity of the band Rush?
It's worth noting here that the band existed for 6 years before Peart joined, so it's unlikely that he owns the rights to the name. On the other hand, most of the band's successes came post-Peart, so he's arguably an essential part of what most people identify as "Rush."
From interviews, it seems like there's no actual barrier to Rush continuing on without Peart, but that there's also no interest in that by the remaining members. They're all technically retirement age by now, and don't seem to have much interest in the rigors of touring, nor have they released any new music since 2012:
"I'd say I can't really tell you much other than that there are zero plans to tour again. As I said earlier, we're very close and talk all the time, but we don't talk about work. We're friends, and we talk about life as friends. I can't really tell you more than that, I'm afraid. I would say there's no chance of seeing Rush on tour again as Alex, Geddy, Neil. But would you see one of us or two of us or three of us? That's possible.
Geddy Lee in Rolling Stone, 2018.