Does there exist a non-electric bike that can go 60mph (95 km/h)?
The problem with going 60 mph (assuming level ground, no wind, etc) is not gearing, which is an easy problem to solve. It's aerodynamics. Aerodynamic drag increases roughly with the square of your speed, and to get up to 60 mph on a conventional bike, you'd need to produce about 4000 W—which some sprinters can do for a few seconds.

So to go anything like that fast, you'd need to be in an aerodynamic HPV. The current hour record for a streamliner HPV is about 90 km, still short of 60 mph, although over sprint distances, HPVs have gotten up to 144 kph. 

However, these would be completely impractical for mass-start events like the Tour de France, so you can't easily extrapolate to what a mass-start HPV race would look like, or what speeds people would achieve, because the bikes would need to be significantly different. Some of these record-attempt HPVs have no windows at all, and rely on cameras and TV screens for outward visibility; the ones with windows have extremely limited visibility. Riders are taped in, and could not easily get out to change a flat, for example. 

Also, drafting plays an extremely important role in (conventional) bike racing. Because the benefits of drafting would be greatly reduced in HPV racing, those races would essentially turn into time trials.