You can prove them wrong but it's like playing chess with a pigeon - you think you're winning, and then the pigeon shits on the board and flies off. Point being that reasoning with these people is a waste of everyone's time because their concerns are not based on reason, so reason can't defuse them. More likely that they'll see you as some sort of government operative, the existence of which PROVES that they're right!
That said - why is this guy crazy, you ask? Well first of all:
There are ways to go about it, but this isn't it...
I'm curious, which ways are that?
Well clearly it's not his way either - it's not like we're all talking about that time we might have had a global positioning system until this guy ruined it, is it? The GPS literally reached "initial operating capacity" (continuous worldwide coverage) on schedule about a year later. I never even heard of this guy until now.
More broadly, sure the GPS is a military invention and is run by the military (though overseen by a committee, and other GNSS are run by different militaries with no particular love for the US). But it didn't take long for it to be opened up to civilians - only one satellite had been launched! By now even Selective Availability has been turned off, and can not be turned on (the new satellites can't do it). Yes, the SA thing was after this guy's rampage, but he's sticking with his story, so I'll count it.
GPS is one of the greatest peacetime things that military technology has ever done (in a long line of technical advances fueled by the military). Think about it - the average person now has at least two devices that know where they are in absolute terms on the Earth's surface, to within a few feet. This has never happened before! People had maps, which are a big enough breakthrough on their own, but are comparatively inaccurate, need to be kept up-to-date, and require some skill to use - and you have to know basically where you are relative to identifiable landmarks in order to use them. Planes can fly routings more precisely, or even directly to the destination, saving fuel and freeing up congested airways. And instrument approaches are now possible to virtually any point on earth - no expensive phased-array radio antenna on the ground to maintain, just define a few points in a database and publish a chart. And all that thousands of years of naval navigation technology (like the sextant, or the clock)? - unnecessary, except perhaps as a backup. Cars with turn-by-turn directions, virtually eliminating the big road atlas or fold-out maps everyone had to have and mess with while driving (and far more accurate that your aunt's "turn right by the, well there used to be a farm there but now it's just a field" directions). Track logs of running and biking sessions to evaluate speed, performance, and trends. Nanosecond-scale timekeeping, allowing for previously-impossible management of the power grid and other distributed systems. Slightly in the future, self-driving cars - and more we haven't even imagined yet. By comparison the military's usage is unsophisticated and unthreatening - it's just a lighter-weight replacement for systems they already had like LORAN and allows bombs to be placed somewhat more accurately and easily than e.g. a laser sight.
Point being, he's arguing (still!) for the destruction of something that would be a far, far greater loss to peaceful civilians around the world than it would be for the military. With all the countries that know how to shoot down satellites nowadays (why did they develop that? hmm), does he think the militaries don't have a contingency that they're more than capable of using?
So, no, this guy doesn't have a point. Protesting the GPS is like protesting computers because they could break German codes or develop artillery tables, protesting the internet because it was designed to enable military communication in the event of a nuclear war, or protesting duct tape because it sealed ammo cases. Yeah military bad rah rah... but come on.