If some billionaire happened to fund the whole thing "just for humanity", do you think the powers that be would even let it happen?
Hard to stop it, short of shooting down the launch vehicle in its first 10-15 minutes of flight.
Rocket launch from US air space? Well, the US could stop it I suppose. But, if that command has to climb a command chain, the craft would already be 100 miles up before a "shoot" command came down. It would be 150 miles up before a US Destroyer with Aegis or similar tech could lock on and launch a kinetic interceptor.
(Assuming that the Air Force's new "micro-shuttle" isn't a god-rod system that can intercept from above... and even if it were, it's on a 2 hour orbit pattern and unlikely to be in position for the 10-15 minutes it takes to achieve orbit.)
By the time a ground-based missile could intercept, the craft would be beyond high earth orbit and on its way. The interceptor missile would run out of fuel in low earth orbit, relying on inertia and pre-plotted trajectory to impact the target. A slight course deviation, and the missile is off target.
That's assuming that the US would shoot DOWN a projectile/rocket going up that shows no signs of being a danger to anyone on the ground.
I doubt the Russians or the Chinese could knock down such a craft, unless the Russians have already put weaponized satellites in space during the height of the Cold War.
If someone held a press conference 1 minute before launch, and 1000 miles away from the secret launch site (ideally a hollowed out volcano), declaring intent to launch and the purpose, I doubt the US Government would have the information assets to stop it before ignition, or chutzpah to forcibly shoot it down in light of the starry-eyed dreams of the billionaire philanthropist whom the public would love so much.
Look how much everyone loves Burt Rutan.
Multiply that by a million-fold if someone funded the next Columbus expedition to the Great Unknown.
You know that Columbus guy, right? Suppose his name will ever die in history?
THAT is your billionaire motivation.