Ignoring the various slander of presidents...
Nobody can "shut down the Internet." By its very nature it's self-repairing. If he shuts down core DNS, there are core DNS in other countries. If he shuts down the border fibers in the US, he turns us into what China dreams of. He'd shut down access to various .coms that only have hosts in the US.
Hell, let him shut it off. Spam would be down a huge amount, and there'd be much more bandwidth left to access The Pirate Bay.
And if they cut off the core routers, the world is fucked, internal networks would operate, but nothing else, and it could take 2 weeks to get everything fully online again