It's unlikely China or Russia, or anyone, has the capability to shoot down GPS sats. A number of countries, the US included, have demonstrated capability to shoot down satellites, low orbit satellites. GPS satellites are in geosynchronous orbit, which is a whole different species of animal. Shooting down an LEO satellite can be done with an SM-3 missile which has a mass of 1.5 tons. That's significantly lower than the mass of a rocket needed to put the satellite in orbit, and that's possible because SM-3 is not designed to reach orbital speeds, only orbital heights. Its primary mission is anti-ballistic missile defense and ballistic missiles can reach thousands of kilometers altitude, but not tens of thousands.

In theory one can design a missile that can reach GSO heights without reaching GSO speeds, but that requires a dedicated project, which would be extremely expensive, and very difficult to keep under wraps. All missile launches are pre-declared to minimize the risk of a nuclear war. You can find a list of all orbital and suborbital launches on wikipedia (e.g. [1]). If anyone tried to test a missile that can reach GSO heights, they couldn't keep it a secret.

One can then try to shoot down GSO satellites with full size rockets, like Falcon Heavy. But there are 31 GPS satellites out there, and 30 Galileo. I doubt either Russia or China (or even America) have 60 full size rockets available to do a blitzkrieg strike against the GPS satellites.

[1] https://en.wikipedia.org/wiki/List_of_spaceflight_launches_i...