No NCR will not fall apart. It is a democracy that isn't dependent on one leader controlling everything. What is happening in the Mojave isn't going on everywhere else in the NCR.
NCR has a large army. It has a system of trains and can out manufacture any of it's current enemies. It has driven the Bos underground. And it's stable.
Sure people in the Mojave complan about the NCR, but when faced with the Legion they can't kiss NCR's ass fast enough.