Yes, I have no doubt they could. But again, why would they? Apple doesn't make most of its software sales by developing and selling their own iOS software. They make their money by developing the iOS platform itself, and then taking a 30% cut of every piece of software that other companies develop for it. With that model, they could conceivably get Nintendo games on their platform without EVER purchasing the company outright. If your apocalyptic vision of Nintendo's future comes to pass, Apple could simply sit back and wait for Nintendo to port their software over of their own free will, and then take their 30% cut. No purchase necessary.
At this juncture, given Apple's business model, purchasing Nintendo would be nothing but a liability. They'd have plenty to lose, and very little to gain. The biggest asset they would acquire from the purchase would be Nintendo's various IP. And Apple isn't really situated to take advantage of that IP. They're best bet would be to license those IP out to other companies. (something that Apple really isn't in the business of doing)
It might give them a temporary bump in stock value, but I don't think it would last long. And it would effectively destroy Nintendo overnight.