For all it worth, right now most anti-satellite weapons are limited to low earth orbit (useful for knocking out satellites looking at stuff you don't want). Most communications satellites will be a much higher altitude.

While there is no fundamental reason why governments couldn't create ASAT weapons to reach higher altitudes, the costs go on significantly. Previous (like cold war era) attempts at high altitude weapons pretty much just strapped a big nuke to a full out launch vehicle. For reference, the weapon used by the US in their ASAT kill in 2006 was made using a modified ballistic defense missile (the SM-3). Your Aegis equipped warships can carry a -lot- of them.

Anyhow, it'd be far easier for governments to attempt to regulate the receivers and transmitters than to shoot down the satellite. Also, it would be easier for them to deny launch privileges than wait till after they launch. They might not own space, but they own their airspace.